YouTube uses a system called Content ID to find copyrighted content. If copyrighted content is uploaded to YouTube and matched against the Content ID database, YouTube will either
- block access to the content
- show adverts with the video for the copyright holder to earn revenue
- allow the video with no advertising
where the action to take is decided by the copyright holder. As such, the copyrighted content is perfectly legal and explicitly permitted for use in the video by the copyright holder (this system even applies to other videos that use the copyrighted content as, for example, background music - personally I had some song parodies recorded over the original backing tracks that were caught by the same system).
Vevo is a specific service for music producers to upload their own music to an "official" channel. If you see a Vevo channel and it's got the "verified" check mark next to it then you can be certain that it's an official channel operated by or on behalf of the original content producers.
For content not present in the Content ID database or identified by the Content ID system, I'm not sure what the legality is. YouTube has typically operated on a policy of "hold no responsibility for illegal content and wait for the copyright holder to request takedown", which they will then honour.
In any case, you cannot be in legal trouble for watching/listening to copyrighted content on YouTube. The legal responsibility falls on YouTube for distributing the content to you, and on the original uploader for copying and uploading the content. AFAIK copyright laws apply to the person producing a copy (either modified or unmodified) of the copyrighted content or providing a copy to someone else, not the person receiving or consuming the content. Watching a video online does not constitute producing a copy (downloading does, however).
It's also very likely that your use of the copyrighted content would fall under fair use (this applies particularly to the "take screenshots of it" part), as it is private and non-commercial use for study/educational purposes.