When you dont log out from sites,what dangers can occur to your account?Who are the people who can take advantage of this(in evil means)?
Totally depends on the implementation of the site's application. Usually they'll time you out.
Is it possible for someone to do something evil to your account? Again, depends on the implementation. If it was encrypted, chances are no. If it used some way to id your session specific to your browser/computer, probably no. But there could be a bug, someone could intercept things and use a man in the middle attack, etc.
NORMALLY they'll just time out your session after a period of inactivity.
This is assuming you're talking about...a web site? If you're talking games you're probably on the wrong forum.