Seems like all the kids these days are using YouTube to listen to songs. It used to be trivially easy to play videos in the background, though, by just starting a video in the YouTube app or in Safari, and then just switching out to another app.
These days, however, the latest iOS version has changed that, and switching out of the YouTube app or Safari with a video playing stops the playback. Never fear, though, there are a couple of workarounds.