Hi @Tease - If you listen to a song or video online, then - depending on the browser/application you use to do that, you may find that it caches that download locally in which case it might download a little bit again but you won't actually download as much if you play it again *immediately*. That cache is cleared automatically though so if you try to listen/watch it later then it is likely to have to download it again. Alternatively there are tools/apps available that enable you to download content off the likes of youtube so you could play them multiple times in the future without having to download them again - however, depending on the content, you need to be careful that you are not breaking any laws in doing so.
"Ninkasi" is mostly right. When you visit any page it involves a certain amount of download. If you refresh a page to look at it again after you have browsed elsewhere the same amount of data will be downloaded. Youtube's data compression is pretty good in that they keep the downloads as sparse as possible but if you watch a film you are still looking at up to half a gig at times. You can use Youtube downloaders in most browsers to obtain the video permanently but if you download copyright material you could land yourself in big trouble. It's not worth the risk.
Depending on the amount of content you're using, you may also want to consider YouTube Red which allows offline playback on mobile devices 🙂