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.
Most databases often indicate whether a journal is peer-reviewed. Some databases may even offer a way to limit results to only peer-reviewed materials. Within Business Source Complete, for example, a search within the Publications provide info about the journal, including if it is peer-reviewed.
Another way is to check the library's catalog, which will indicate whether a journal is peer-reviewed (example below). Search for the journal name, and under the name of the journal it will indicate if it is peer-reviewed.
Another way is to visit the journal or publisher's web site. Publishers usually indicate if a journal is peer-reviewed.
Just because a journal is deemed peer-reviewed does not mean that every article within that journal is peer-reviewed. For example, book reviews, editorials, and some other sections of a journal do not qualify as peer-reviewed.