Quake on Android
If you want to play Quake on your Android smartphone you have several choices.
glesquake
glesquake has no audio and you need hardware buttons. It seems to be a typical "curiosity" port. It seems technically impressive and well-planned but not tailored to user-friendliness. It is not actively developed anymore. You can read more about it at http://grammerjack.blogspot.com/2009/10/gles-quake-port-of-quake-to-android.html
glesquake is available for easy installation through the free software repository F-Droid: http://f-droid.org/repository/browse/?fdfilter=quake&fdid=com.android.quake
quake android
https://code.google.com/p/quake-android/ seems to be taking the glesquake codebase and adds touch controls to it. It does not offer an .apk outside the Google Play store.
FTEQW/FTEDroid
FTEQW is a cross-platform Quake engine that runs even on your electric toothbrush. It has an Android port as well. You can directly download the .apk at http://triptohell.info/moodles/android/FTEDroid.apk (this download is updated regularly, you will be running the bleeding edge of software development).
With FTEDroid you can play singleplayer, play or spectate multiplayer (both Quakeworld and Netquake, QTV works very well), mods, everything really. Put your pak0.pak and pak1.pak into /sdcard/fte/id1/ . On a Nexus One timedemo demo1
gives 18fps.
You can find some more usage info and discussion in http://www.quakeworld.nu/forum/topic/5969/quake-ported-to-windows-phone-8 (regardless of the title).
There might be more engines out there, so please add them to this page. Do not promote engines where the source was/is not released.