How do you log into facebook on another users computer?

you should be able to the way someone would if a computer owner logged onto his/her facebook account. but what about a computer at a library. I can't access my account from it. It says that someone tried to hack it. But, both times i tried it showed it so I know that it's not real. I have an active account, but I can only access it if I use a specific computer, so do I make another account at the library or what? All answers appreciated Thanks. can't find the answer anywhere.