Peer Review is a process that journals use to ensure the articles they publish represent the best scholarship currently available. When an article is submitted to a peer reviewed journal, the editors send it out to other scholars in the same field (the author's peers) to get their opinion on the quality of the scholarship, its relevance to the field, its appropriateness for the journal, etc.
Publications that don't use peer review (Time,Cosmo, Salon) just rely on the judgement of the editors whether an article is up to snuff or not. That's why you can't count on them for solid, scientific scholarship (Life Science Library, University of Texas at Austin).
Still confused? The videos below, produced by Western College in Canada and North Carolina State University, and the information graphic at right outline the process visually.
Video viewing tip: Click on the button with the four corner brackets to view the videos fullscreen.