After config Jenkins to compile and test Android project, it could be configured to save the apk file generated for each compilation.
1. Go to the configuration of the job and under "Post-build Actions", click on "Add post-build action"
2. Select "Archive the artifacts"
3. In the folder Files to archive, enter the path to the APK file. Usually it is under the folder bin and it is the name of the project and "-debug.apk".
For example, for the project "AutoCompleteTextViewHack", the apk is under "bin/AutoCompleteTextViewHack-debug.apk".
If you don't know which one is, compile the project and then, on the main page of the job click on "Workspace" and browser in the bin folder to find the apk you want to save.