You plagiarize when you take someone's ideas or words and pass them off as your own. It's a fancy word for copying. When you write your first research paper, your teacher will probably spend some time explaining why you shouldn't plagiarize — and the consequences if you do.
It's not illegal to plagiarize but it's morally wrong, and it might just get you tossed out of school. If you're writing something and include someone else's words or ideas without attribution, you are plagiarizing. The origins of the word are said to mean "one who kidnaps the child of another" — which certainly gives a sense of the gravity of the charge. After all, some writers do consider their words to be their babies.