In a non-refereed magazine or journal, the decision regarding which articles to publish are made by their staff -- perhaps an editor or editorial board.
A peer-reviewed or refereed journal is one in which the articles are subjected to an external review process to assist in determining whether or not it should be published. Specialists or scholars in the same discipline review a draft of the article to evaluate the quality of scholarship, clarity & soundness of the research & conclusions, its contribution to the field, etc. The reviewers might suggest modifications to the article prior to publication. Often a "blind" process is used wherein the article's author is not identified to the reviewers to aid in maintaining their objectivity.
Scholarly academic journals are used to share the results of research with other researchers in that field of study. The articles typically include bibliographies with citations to sources. Not every scholarly journal makes use of the peer-review process, but many online databases treat the concepts as synonymous.
Many professors who instruct their students that they must use a certain number of peer-reviewed articles in their research just want to make sure that the students research in the scholarly literature rather than basing their writing entirely on free websites, newspaper articles, popular magazines, or trade journals. The student might want to verify with the professor how strict a definition of "peer-reviewed" is intended in case articles identified as scholarly in a library database are acceptable as well.
Search the Ulrich's Global Serials Directory (formerly Ulrich's International Periodicals Directory) by journal title to determine whether or not it is refereed (peer reviewed). In the search results list, refereed journals have a small picture of a referee's shirt with black & white vertical stripes. The full descriptions of individual titles have a refereed field identifying "yes" or "no".
Ulrich's does NOT index journal articles, but it will identify which databases index the articles within the journal and you can search those databases for your subject.