Peer review refers to an editorial process in place at many academic journals. After an article is submitted, it is sent out to a few people with some degree of knowledge about the article's topic. These might be university professors or other professionals in the field (a medical journal might use doctors among its peer reviewers). These reviewers read the article and make recommendations; they might say it is ready for publication or reject it, or they might say that certain parts of it need to be revised.
So by the time an article is published in a peer-reviewed journal, it has been subject to this process—people who know something about the topic have judged it to be solid scholarship. Peer review is one of the ways that information can become credible. That's why a lot of your professors might be asking you to include peer-reviewed articles in your bibliography.
Here's a nice video on the peer-review process produced by the North Carolina State Libraries: