Peer review is the process by which articles or other works are critiqued before they are published. Authors send articles to an editor, who decides whether the work should be forwarded to reviewers for the journal. The most stringent form is anonymous or blind review, where neither the author nor the reviewers know whose work is being examined by whom. This helps reduce bias.
Reviewers are usually well-published researchers and experts. They return the articles to the editor with remarks and recommendations-- usually publish as is (rare), publish if edited or changed in specific ways, or don't publish. Editors most often go with the recommendation of the majority of the reviewers.
The process is intended to improve the studies published-- more eyes on a project, and one's reputation on the line with peers, tends to improve the quality of what's published. There are cases where it hasn't worked, and critics of the cycle, but it is the best system that has been developed to this point.
Peer-reviewed or referreed or scholarly are often descriptions used interchangably for reputable journals. Not all scholarly journals are peer-reviewed, but many are.