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. --University of Texas at Austin
A primary difference between scholarly journals and other types of journals and magazines is that articles in these journals undergo a "peer review" process before they are published. What does this mean?
Peer review is the process by which an author's peers, recognized researchers in the field, read and evaluate a paper (article) submitted for publication and recommend whether the paper should be published, revised, or rejected.
Peer review is a widely accepted indicator of quality scholarship in a discipline or field. Articles accepted for publication through a peer review process meet the discipline's expected standards of expertise.
Peer-reviewed (or refereed) journals are scholarly journals that only publish articles that have passed through this review process. --San Diego State University Library