Let's say you're listening to a 2-hour concert while reading a book. Since you only care about the audio, YouTube wastes bandwidth by streaming the entire video. YouTube now uses adaptive streaming (DASH) and there are separate streams for audio and video.
I mentioned in a recent post that the mobile YouTube app for Android downloads only the audio files when playing music videos in the background. The desktop site doesn't do this, but you can save bandwidth by switching to the lowest quality option that's available: 144p. Click the player's settings button (the wheel icon) and pick 144p from the "quality" dropdown.
It may not seem obvious, but YouTube uses exactly the same audio stream, whether you're watching 144p, 240p, 360p, 480p, 720p or 1080p videos. You can check this by right clicking the player, selecting "stats for nerds" and watching the DASH section when switching to other resolutions. The first numerical value is the ID (itag) of the video stream, while the second one is the ID of the audio stream. You'll notice that the audio ID is the same: 140.
Here's an example: the 2-hour David Gilmour concert from the screenshot above uses 237MB for the 360p stream and 94MB for 144p stream, while the audio stream uses 111MB. That means you can save more than 140MB by switching from 360p to 144p.
Please note that this only works if you're using the Flash player or you're using YouTube's HTML5 player in a browser that supports Media Source Extensions (Chrome, Opera, Safari, IE11).