Peer-reviewed journal articles have gone through a special review process (peer review, naturally). This means scholars of a discipline have read an article carefully and determined that it makes a worthwhile contribution and was researched correctly. This is why instructors prefer you to use peer-reviewed articles -- it guarantees that your sources are good ones.
When your instructor asks you to use peer-reviewed articles, a database is where you'll find them. There are databases for any subject you can think of, and they'll usually let you limit your search (for example only to peer-reviewed articles, or to articles from the past five years).
The databases on this list are multidisciplinary -- they focus on research from all areas of knowledge. We also have subject-specific databases if you know exactly which field your topic falls into (for example, if you're writing about sociology or nursing).
