A
It's an acronym for Android application package. It is the file format used to install application software on your Android phone. They are usually ZIP file formatted packages, based on the JAR file format, with an .apk file extension.
In layman terms, think of it like a setup.exe in Windows or a .dmg file for OS X which let you install software on your device.
If you're thinking of downloading these, make sure that the source is trust-worthy. Or else it's better to stick with Google's Play Store.
https://en.wikipedia.org/wiki/Android_application_package
Edit : B.Diddy beat me to it!