I was working on an Android project using Xamarin in Visual Studio 2012. I recently upgraded from an HDD to a SSD so I reinstalled Windows and all of my programs.
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I solved the issue. Somehow the application still existed on my device even though it did not show up in my applications and I needed to remove it.
The solution was to find an old APK that I had emailed and I installed that. Then I uninstalled the application and deployed it again from Visual Studio. This time it worked.
You can read more here: https://forums.xamarin.com/discussion/8501/install-failed-update-incompatible
EDIT (March 13th, 2017):
Seems a lot of people found this answer useful so I decided I should update it with an alternate method to uninstall the app if you don't have access to an old version.
As Atul Chaudhary and Nestel mentioned, you can open up your adb command prompt and run adb uninstall . If there are any remnants of your application which remain this should get rid of them.