What Does "Peer Reviewed" Mean?
When a journal is peer-reviewed (also called "Refereed"), it means that all articles submitted for publication have gone through a rigorous evaluation. To ensure that each article meets the highest standards of scholarship, it is evaluated by the editor(s) of the journal. This is sometimes called "internal review." In some cases multiple in-house research editors will evaluate the scholarly strength of the article before even considering it for further peer-review.
The editor(s) then enlists the services of other scholars in the same field as the manuscript's author--in other words, that author's peers. This peer-review process is sometimes called "external review." These peer scholars offer their view on the quality of the article and its research. How appropriate and exacting was the research method? Were the results presented in the best way possible? Was the literature review thorough? Does the article make a significant contribution to the scholarship of that field? Does it meet the scope of this particular journal? There are many criteria for judgement!
The exact peer-review process will vary between publishers. Additionally, the whole value of peer review, as it now exists, is often hotly debated. Some believe that the Open Access (OA) movement of publishing research on the web and inviting scrutiny and comment will eventually eliminate the need for publisher driven peer-review.
However, as of now, peer-reviewed journal literature is still considered the highest form of scholarship. Also, your professors will likely say that they want you to use peer-reviewed articles in your paper--sometimes exclusively.