In simplest terms, plagiarism is the use of the work of someone else and claiming it to be yours. While this sounds like something you would obviously never do, there are many gray areas where people accidentally plagiarize when they don't mean to. Usually, this occurs when someone forgets to cite a piece of information in a paper that is either a direct quote or a paraphrasing from an author. The most important thing about avoiding plagiarism is that you can never be too safe. When in doubt, cite. When not in doubt, but it's clearly not your own interpretation/analysis, cite. Even if a sentence is largely your analysis of one fact, if you introduce that fact in that same sentence, cite the fact at the end of the sentence.
Unless you have achieved a level of mastery of your subject (likely no one who is pursuing an undergraduate degree), most of the information you use in a paper did not spring forth from your brain with no original source. With the exception of basic facts that are considered "common knowledge", you should cite where you got your facts about mitochondrial DNA, the Hapsburg Empire, symbolism in Emily Dickinson's poetry, Jung's theory of the collective unconscious, or whatever other subject on which you might be writing.
It is also important to be aware of self-plagiarism. That is, you shouldn't lift whole paragraphs from one paper you wrote into another, without citing yourself (after first asking your professor if this is an acceptable practice for the assignment).
Plagiarism is one of the most serious violations of academic or professional integrity a person can make. Those who plagiarize are often caught (it's hard to fool professors who have read widely across their respective fields) and the penalties are severe and last a lot longer than merely receiving an 'F' grade. Plagiarism is not a crime per se, but it can lead to possible accusations of copyright violation which is a crime.