Basically, YouTube doesn't go on search-and-destroy missions for
videos they think may be in violation of copyright: considering that
in the time it took you to write your post, enough videos were
uploaded to YouTube to keep you watching for more than a day, it
simply wouldn't be possible.
What happens is this: If somebody sees a video that violates their
copyright, they have the option, if they wish, of ordering YouTube to
remove or mute the video. The principle is that only the copyright
owner has that right, and YouTube doesn't want to risk taking the law
into their own hands and pre-emptively removing videos that may be
perfectly illegal, purely on suspicion.
This does mean that some videos escape detection. There may be
hundreds still up, but you can pretty much be certain that thousands
have been removed.
In this sense, YouTube is very much between a rock and a hard place.
Viacom are hoping to sue YouTube for a billion dollars, claiming that
the law says they must remove all infringing copyright immediately.
YouTube's argument is that they cannot know a video is illegal until
the copyright owner tells them, and that what Viacom is asking them to
do is unreasonable. Imagine if every day, somebody at YouTube had to
send millions of e-mails saying, "We think this video infringes your
copyright, what do you want us to do?" -- remembering that before they
can do that, they actually have to work out who the copyright owner is.