Wake Android Device up

To wake up the screen:

PowerManager pm = (PowerManager) getApplicationContext().getSystemService(Context.POWER_SERVICE);
WakeLock wakeLock = pm.newWakeLock((PowerManager.SCREEN_BRIGHT_WAKE_LOCK | PowerManager.FULL_WAKE_LOCK | PowerManager.ACQUIRE_CAUSES_WAKEUP), "TAG");
wakeLock.acquire();

To release the screen lock:

KeyguardManager keyguardManager = (KeyguardManager) getApplicationContext().getSystemService(Context.KEYGUARD_SERVICE); 
KeyguardLock keyguardLock = keyguardManager.newKeyguardLock("TAG");
keyguardLock.disableKeyguard();

And the manifest needs to contain:

<uses-permission android:name="android.permission.WAKE_LOCK" />
<uses-permission android:name="android.permission.DISABLE_KEYGUARD" />

For more details about PowerManager, refer to the API documentation: http://developer.android.com/reference/android/os/PowerManager.html

EDIT: this answer is reported as deprecated.

Best is to use some appropriate combination of these window flags:

http://developer.android.com/reference/android/view/WindowManager.LayoutParams.html#FLAG_DISMISS_KEYGUARD
http://developer.android.com/reference/android/view/WindowManager.LayoutParams.html#FLAG_SHOW_WHEN_LOCKED
http://developer.android.com/reference/android/view/WindowManager.LayoutParams.html#FLAG_KEEP_SCREEN_ON
http://developer.android.com/reference/android/view/WindowManager.LayoutParams.html#FLAG_TURN_SCREEN_ON

If you want to run on older versions of the platform that don't support the desired flag(s), you can directly use wake locks and keyguard locks... but that path is fraught with peril.

ONE IMPORTANT NOTE: Your activity must be full screen in order for the above flag combination to work. In my app I tried to use these flags with an activity which is not full screen (Dialog Theme) and it didn't work. After looking at the documentation I found that these flags require the window to be a full screen window.

I found a way and it is not that complex... works on any API version.

You need to use PowerManager.userActivity(l, false) method and register your activity as broadcast received for SCREEN_OFF intent:

In your actiivity OnCreate put something like:

mReceiver = new BroadcastReceiver() {
    @Override
    public void onReceive(Context context, Intent intent) {
        Log.v(TAG, "Screen OFF onReceive()");
        screenOFFHandler.sendEmptyMessageDelayed(0, 2000L);
    }
};

It will kick off the handler after 2 seconds of Screen Off event.

Register receiver in your onResume() method:

IntentFilter filter = new IntentFilter(Intent.ACTION_SCREEN_OFF);
registerReceiver(mReceiver, filter);
Log.i(TAG, "broadcast receiver registered!");

Create a handler like the one below:

private Handler screenOFFHandler = new Handler() {

    @Override
    public void handleMessage(Message msg) {

        super.handleMessage(msg);
        // do something
        // wake up phone
        Log.i(TAG, "ake up the phone and disable keyguard");
        PowerManager powerManager = (PowerManager) YourActivityName.this
                .getSystemService(Context.POWER_SERVICE);
        long l = SystemClock.uptimeMillis();
        powerManager.userActivity(l, false);//false will bring the screen back as bright as it was, true - will dim it
    }
};

Request permission in your manifest file:

<uses-permission android:name="android.permission.WAKE_LOCK" />

Do not forget to unregister broadcast receiver when you are done. You may do that in onDestroy() for example (which is not guaranteed)

unregisterReceiver(mReceiver);
Log.i(TAG, "broadcast UNregistred!");