Peer-reviewed research has gone through a refereeing process (like a sports referee). Through the peer review process, a scholarly work is checked by a group of experts in the same field to make sure it meets the professional standards before it is published. It is largely used with scholarly research journals (academic journals) to help ensure that published articles represent the best scholarship that is currently available.
Publications that do not use peer-review, such as Time, Discover, Newsweek, and U.S. News, rely on the judgment of the editors as to whether an article is quality material or not. Articles are not as rigorously reviewed because these publications do not rely on solid, scientific scholarship.
How do I know if a journal is peer-reviewed?
Most article databases, including the library catalog, have an option to limit results to peer review. If there is not an option to limit, try adding "peer review" as a search term. Many databases like Academic Search Complete (EBSCO) indicate whether an article is peer reviewed. 
Peer-Review in Three Minutes
This three minute video describes and discusses the importance of peer-review and its process.
(NCSU video, 3:15 minutes)