Is there a way to run Android on a Raspberry Pi? Or at least run Android apps? I know there are ways to run Android apps in Chrome, but does this work on the Pi?
Check out the community driven Razdroid project that is however - as of today - far from being able to provide a finished version of Android for the RPi. It is stated that they mainly suffer from a proper port of hardware acceleration drivers for the Broadcom SoC used on the RPi. So - again as of today - depending on your usage the RPi might not be the best platform to run Android at a high performance.
Update May 2016
Arstechnica found some evidence of future Android support for the Pi3:
In Google's Android Open Source Project (AOSP) repository, a new device tree recently popped up for the Raspberry Pi 3. [...] For now, the Ri 3 [sic] device tree is empty with only the comment "initial empty repository" accompanying it. The repository should soon start to fill with code, though.
Android Pi Wikia is a forum for an unofficial version of Android for Pi