This table courtesy of the University of Texas San Antonio Library
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. The review process helps ensure that the published articles reflect solid scholarship in the field of study.
Publications that don't use peer review (Time, Cosmo, New Scientist) just rely on the judgment 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.
If you are not certain, you should look at the journal's Editorial Statement or Instructions to Authors for a description of the process authors are required to go through. Another good source for this information is the journal's website. You can also ask your professor or a librarian for help.
Many of our databases, including Academic Search Complete, allow you to refine your search to limit results to scholarly (also called peer-reviewed or refereed) articles. Some others, for example JSTOR and Project Muse, contain only scholarly material.
Depending on the nature of your research, both scholarly journals and popular magazines can be valuable sources.
If you are unable to find enough information to determine if a journal is scholarly, ask a librarian for help.