Uso de NanoHTTPD en Android

El código fuente bastante bueno se puede encontrar aquí: https://github.com/Teaonly/android-eye

Carpeta de Chceck assets donde se almacenan los archivos html y JavaScript https://github.com/Teaonly/android-eye/tree/master/assets

TeaServer-implementación del servidor https://github.com/Teaonly/android-eye/blob/master/src/teaonly/droideye/TeaServer.java

MainActivity-server inicialización https://github.com/Teaonly/android-eye/blob/master/src/teaonly/droideye/MainActivity.java

Actualizado WebServer clase (ver respuesta de rendon) que funciona con la versión actual de NanoHTTPD:

private class WebServer extends NanoHTTPD {

    public WebServer() {
        super(8080);
    }

    @Override
    public Response serve(IHTTPSession session) {
        String answer = "";
        try {
            // Open file from SD Card
            File root = Environment.getExternalStorageDirectory();
            FileReader index = new FileReader(root.getAbsolutePath() +
                    "/www/index.html");
            BufferedReader reader = new BufferedReader(index);
            String line = "";
            while ((line = reader.readLine()) != null) {
                answer += line;
            }
            reader.close();

        } catch(IOException ioe) {
            Log.w("Httpd", ioe.toString());
        }

        return newFixedLengthResponse(answer);
    }

}

Prueba esto... Crear 2 paquetes (actividad, util), solo para la organización En actividad crea la clase MainActivity.java en util crear la clase AndroidWebServer.java

package awserverfatepi.com.activity;
import android.content.BroadcastReceiver;
import android.content.Context;
import android.content.DialogInterface;
import android.content.Intent;
import android.content.IntentFilter;
import android.net.ConnectivityManager;
import android.net.NetworkInfo;
import android.net.wifi.WifiManager;
import android.support.design.widget.CoordinatorLayout;
import android.support.design.widget.FloatingActionButton;
import android.support.design.widget.Snackbar;
import android.support.v4.content.ContextCompat;
import android.support.v7.app.AlertDialog;
import android.support.v7.app.AppCompatActivity;
import android.os.Bundle;
import android.view.KeyEvent;
import android.view.View;
import android.widget.EditText;
import android.widget.TextView;

import awserverfatepi.com.R;
import awserverfatepi.com.util.AndroidWebServer;

public class MainActivity extends AppCompatActivity {

    private static final int DEFAULT_PORT = 8080;

    private AndroidWebServer androidWebServer;
    private BroadcastReceiver broadcastReceiverNetworkState;
    private static boolean isStarted = false;

    private CoordinatorLayout coordinatorLayout;
    private EditText editTextPort;
    private FloatingActionButton floatingActionButtonOnOff;
    private View textViewMessage;
    private TextView textViewIpAccess;

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);

        initGui();
        setIpAccess();
        floatingActionButtonOnOff = (FloatingActionButton) findViewById(R.id.floatingActionButtonOnOff);
        floatingActionButtonOnOff.setOnClickListener(new View.OnClickListener() {
            @Override
            public void onClick(View v) {
                if (isConnectedInWifi()) {
                    if (!isStarted && startAndroidWebServer()) {
                        isStarted = true;
                        textViewMessage.setVisibility(View.VISIBLE);
                        floatingActionButtonOnOff.setBackgroundTintList(ContextCompat.getColorStateList(MainActivity.this,
                                R.color.colorGreen));
                        editTextPort.setEnabled(false);
                    } else if (stopAndroidWebServer()) {
                        isStarted = false;
                        textViewMessage.setVisibility(View.INVISIBLE);
                        floatingActionButtonOnOff.setBackgroundTintList(ContextCompat.getColorStateList(MainActivity.this,
                                R.color.colorRed));
                        editTextPort.setEnabled(true);
                    }
                } else {
                    Snackbar.make(coordinatorLayout, getString(R.string.wifi_message), Snackbar.LENGTH_LONG).show();
                }
            }
        });

        initBroadcastReceiverNetworkStateChanged();
    }

    private void initGui() {
        coordinatorLayout = (CoordinatorLayout) findViewById(R.id.coordinatorLayout);
        editTextPort = (EditText) findViewById(R.id.editTextPort);
        textViewMessage = findViewById(R.id.textViewMessage);
        textViewIpAccess = (TextView) findViewById(R.id.textViewIpAccess);
    }

    private boolean startAndroidWebServer() {
        if (!isStarted) {
            int port = getPortFromEditText();
            try {
                if (port == 0) {
                    throw new Exception();
                }
                androidWebServer = new AndroidWebServer(port);
                androidWebServer.start();
                return true;
            } catch (Exception e) {
                e.printStackTrace();
                Snackbar.make(coordinatorLayout, "A porta " + port + " não está funcionando, por favor altere para outra no intervalo" +
                        " entre 1000 e 9999.", Snackbar.LENGTH_LONG).show();
            }
        }
        return false;
    }

    private boolean stopAndroidWebServer() {
        if (isStarted && androidWebServer != null) {
            androidWebServer.stop();
            return true;
        }
        return false;
    }

    private void setIpAccess() {
        textViewIpAccess.setText(getIpAccess());
    }

    private void initBroadcastReceiverNetworkStateChanged() {
        final IntentFilter filters = new IntentFilter();
        filters.addAction("android.net.wifi.WIFI_STATE_CHANGED");
        filters.addAction("android.net.wifi.STATE_CHANGE");
        broadcastReceiverNetworkState = new BroadcastReceiver() {
            @Override
            public void onReceive(Context context, Intent intent) {
                setIpAccess();
            }
        };
        super.registerReceiver(broadcastReceiverNetworkState, filters);
    }

    private String getIpAccess() {
        WifiManager wifiManager = (WifiManager) getSystemService(WIFI_SERVICE);
        int ipAddress = wifiManager.getConnectionInfo().getIpAddress();
        final String formatedIpAddress = String.format("%d.%d.%d.%d", (ipAddress & 0xff), (ipAddress >> 8 & 0xff),
                (ipAddress >> 16 & 0xff), (ipAddress >> 24 & 0xff));
        return "http://" + formatedIpAddress + ":";
    }

    private int getPortFromEditText() {
        String valueEditText = editTextPort.getText().toString();
        return (valueEditText.length() > 0) ? Integer.parseInt(valueEditText) : DEFAULT_PORT;
    }

    public boolean isConnectedInWifi() {
        WifiManager wifiManager = (WifiManager) getSystemService(Context.WIFI_SERVICE);
        NetworkInfo networkInfo = ((ConnectivityManager) getSystemService(Context.CONNECTIVITY_SERVICE)).getActiveNetworkInfo();
        if (networkInfo != null && networkInfo.isAvailable() && networkInfo.isConnected()
                && wifiManager.isWifiEnabled() && networkInfo.getTypeName().equals("WIFI")) {
            return true;
        }
        return false;
    }

    public boolean onKeyDown(int keyCode, KeyEvent evt) {
        if (keyCode == KeyEvent.KEYCODE_BACK) {
            if (isStarted) {
                new AlertDialog.Builder(this)
                        .setTitle(R.string.warning)
                        .setMessage(R.string.dialog_exit_message)
                        .setPositiveButton(getResources().getString(android.R.string.ok), new DialogInterface.OnClickListener() {
                            public void onClick(DialogInterface dialog, int id) {
                                finish();
                            }
                        })
                        .setNegativeButton(getResources().getString(android.R.string.cancel), null)
                        .show();
            } else {
                finish();
            }
            return true;
        }
        return false;
    }

    @Override
    protected void onDestroy() {
        super.onDestroy();
        stopAndroidWebServer();
        isStarted = false;
        if (broadcastReceiverNetworkState != null) {
            unregisterReceiver(broadcastReceiverNetworkState);
        }
    }

}

En el AndroidWebserver.java

package awserverfatepi.com.util;
import java.util.Map;
import fi.iki.elonen.NanoHTTPD;

public class AndroidWebServer extends NanoHTTPD {

    public AndroidWebServer(int port) {
        super(port);
    }

    public AndroidWebServer(String hostname, int port) {
        super(hostname, port);
    }

    @Override
    public Response serve(IHTTPSession session) {
        String msg = "<html><body><h1>Hello World</h1>\n";
        Map<String, String> parms = session.getParms();
        if (parms.get("username") == null) {
            msg += "<form action='?' method='get'>\n  <p>Seu nome: <input type='text' name='username'></p>\n" + "</form>\n";
        } else {
            msg += "<p>Hello, " + parms.get("username") + "!</p>";
        }
        return newFixedLengthResponse( msg + "</body></html>\n" );
    }
}

No olvides el Manifiesto.xml

<uses-permission android:name="android.permission.INTERNET" />
<uses-permission android:name="android.permission.ACCESS_WIFI_STATE" />
<uses-permission android:name="android.permission.ACCESS_NETWORK_STATE" />
<uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE" />

Y último

Activity_main.xml

<?xml version="1.0" encoding="utf-8"?>
<android.support.design.widget.CoordinatorLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    android:id="@+id/coordinatorLayout"
    android:layout_width="match_parent"
    android:layout_height="match_parent">

    <LinearLayout
        android:layout_width="match_parent"
        android:layout_height="match_parent"
        android:orientation="vertical">

        <ImageView
            android:layout_width="match_parent"
            android:layout_height="150dp"
            android:scaleType="centerCrop"
            android:src="@drawable/header" />

        <LinearLayout
            android:layout_width="match_parent"
            android:layout_height="0dp"
            android:layout_weight="1"
            android:background="@color/colorPrimaryLight"
            android:gravity="center"
            android:orientation="vertical">

            <LinearLayout
                android:layout_width="wrap_content"
                android:layout_height="wrap_content"
                android:orientation="horizontal">

                <TextView
                    android:id="@+id/textViewIpAccess"
                    android:layout_width="wrap_content"
                    android:layout_height="wrap_content"
                    android:text="http://000.000.000.000:"
                    android:textColor="@android:color/white"
                    android:textSize="20sp"
                    android:textStyle="bold" />

                <EditText
                    android:id="@+id/editTextPort"
                    android:layout_width="60dp"
                    android:layout_height="wrap_content"
                    android:gravity="center"
                    android:hint="8080"
                    android:inputType="numberDecimal"
                    android:maxLength="4"
                    android:text="8080"
                    android:textColor="@android:color/white"
                    android:textSize="20sp"
                    android:textStyle="bold" />

            </LinearLayout>

            <TextView
                android:id="@+id/textViewMessage"
                android:layout_width="wrap_content"
                android:layout_height="wrap_content"
                android:layout_marginLeft="50dp"
                android:layout_marginRight="50dp"
                android:layout_marginTop="50dp"
                android:gravity="center"
                android:text="@string/message"
                android:textColor="@android:color/white"
                android:textSize="18sp"
                android:visibility="invisible" />

        </LinearLayout>

    </LinearLayout>

    <android.support.design.widget.FloatingActionButton
        android:id="@+id/floatingActionButtonOnOff"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:layout_gravity="end|bottom"
        android:layout_margin="16dp"
        android:elevation="4dp"
        android:src="@drawable/on_btn"
        app:backgroundTint="@color/colorRed" />

</android.support.design.widget.CoordinatorLayout>

Eche un vistazo a cómo sirvo archivos HTML y otros tipos de archivos también. Tengo una clase AndroidWebServer que extiende la clase Nanohttpd. Este es mi método de respuesta -->

    public Response serve(IHTTPSession session) {
    String uri=session.getUri();
    String msg = "<html><body><h1>Hello server</h1>\n";

    File [] arrayfile;

    int i=0;

    try{
        session.parseBody(new HashMap<String, String>());
    }catch (ResponseException | IOException r){
        r.printStackTrace();
    }


    Map<String, String> parms = session.getParms();
    if (parms.get("username") == null) {
        msg += "<form action='?' method='get'>\n  <p>Your name: <input type='text' name='username'></p>\n" + "</form>\n";
    } else {
        msg += "<p>Hello, " + parms.get("username") + "!</p>";
    }
    msg += "<br><br><a href='/Open_rap'>Open Image of Lionel Messi</a><br><br>";
    msg += "<br><br><a href='/files'>Browse Files</a><br><br>";
    msg += "<br><br><a href='/getmethod'>GET METHOD OPERATION</a><br><br>";
    msg += "<br><br><a href='/postmethod'>POST METHOD OPERATION</a><br><br>";
    msg += "<br><br><a href='/jquery'>JQUERY OPERATION</a><br><br>";
    if(uri.equals("/hello")){
            String response="Hello World";
            return  newFixedLengthResponse(response);
    }
    else if(uri.equals("/getmethod")){
        String html="<html><head><h1>Welcome to the Form</h1><head/><body>";

        if(parms.get("name")==null){
            html +="<form action='' method='get'> \n " +
                    "<p>Enter Your Name:</p> <input type='text' name='name'>" +
                    "</form>" +
                    "</body>";
        }
        else{
            html +="<p>Hello Mr. "+ parms.get("name") +"</p>"+
                    "</body> ";
        }

        html +="</html>";
        return newFixedLengthResponse(html);

    }
    else if(uri.equals("/postmethod")){
        String html="<html><head><h1>Welcome to the Form</h1><head/><body>";
        Map<String, String> files = new HashMap<String, String>();
        Method method = session.getMethod();
        String postParameter="";


        html +="<form action='' method='post'> \n " +
                "<p>Enter Your Name:</p> <input type='text' name='name'>" +
                "</form>";

        if (Method.POST.equals(method) || Method.PUT.equals(method)) {
            try {
                session.parseBody(files);
            } catch (IOException ioe) {
                try {
                   // return newFixedLengthResponse(Response.Status.INTERNAL_ERROR, MIME_PLAINTEXT, "SERVER INTERNAL ERROR: IOException: " + ioe.getMessage());
                } catch (Exception e) {
                    e.printStackTrace();
                    Log.d("Exception", e.getMessage());
                }
            } catch (ResponseException re) {
                try {
                   // return newFixedLengthResponse(re.getStatus(), MIME_PLAINTEXT, re.getMessage());
                } catch (Exception e) {
                    e.printStackTrace();
                    Log.d("Exception", re.getMessage());
                }
            }
        }
        html +="</body></html>";
        String postBody = session.getQueryParameterString();
        postParameter = session.getParms().get("name");
        Log.d("Postbody",postBody+"\n"+postParameter);
        if(postParameter!=null){
            String html1="<html><head><h1>"+ postParameter +"</h1><head></html>";
            return newFixedLengthResponse(html1);
        }
        return newFixedLengthResponse(Response.Status.OK,"text/html",html);
    }

    else if(uri.equals("/Open_rap")){

        File root= Environment.getExternalStorageDirectory();
        FileInputStream fis = null;
        File file = new File(root.getAbsolutePath() + "/www/messi.jpg");
        Log.d("Path",root.getAbsolutePath());
        try{
            if(file.exists())
            {
                fis = new FileInputStream(file);

            }
            else
                Log.d("FOF :", "File Not exists:");
        }catch (FileNotFoundException e)
        {
            e.printStackTrace();
        }


        return newFixedLengthResponse(Response.Status.OK,"image/jpeg",fis, file.length() );
    }
  else {
        return newFixedLengthResponse(msg + "</body></html>\n");
    }

}

Como puedes ver, he implementado el método GET y POST. Puedes encontrar cada uno en esta parte del código uri.equals("/getmethod") y uri.equals("/getmethod").
Además, puede ver la parte {>uri.equals("/openrap"), aquí estoy sirviendo un archivo JPG al navegador del cliente y la imagen está presente en el directorio interno en la carpeta /www/.
Hazme un ping si tienes cualquier duda.