Once upon a time, Kodi was known as XBMC, and it was developed for hacked Xboxes. This open source media hub went on to run on PCs and mobile devices, but it hasn't been deployed in the Play Store until now. There are some not-so-secret Google+ communities for beta and alpha versions of Kodi, and you can join them right now.
The team behind Kodi hasn't publicized the existence of the app on the Play Store, but neither has it made the groups invite only. The devs were apparently hoping to keep it on the down-low until all the kinks were worked out (sorry, guys). If you're cool with a few bugs, you might want to join up. Having the app in the Play Store instead of sideloaded means you'll get automatic updates as new versions are released.
The process to get access should be familiar to anyone who has participated in a Google Play testing program before. Simply choose either the alpha (Isengard) or beta (Helix) group based on your tolerance for bugs. Join that, opt into the test, and download the app. Then settle in and enjoy your media in a rad way.