Specifically, the YouTube app will stop working on first and second generation Apple TV devices, as well as iOS devices running version 6 or older.

Google says that the shut down of older YouTube apps is being done to accommodate new API changes and features for newer version:

As we upgrade the YouTube Data API to bring more features, we'll begin shutting down the old version on April 20, 2015. This will result in the current YouTube app not working on certain device models from 2012 and older.

Google says that the shutdown will begin today, April 20, and older versions will no longer be supported "after April." Third generation Apple TVs and iOS devices running version 7 or 8 will remain unaffected. Other devices losing support after April also include older smart TVs, Blu-Ray players, and devices running Google TV versions 1 and 2.

Source: Google; Via: MacRumors