A periodical is any publication that comes out regularly or occasionally (periodically). Sports Illustrated, The Journal of Anthropological Research, The World Almanac, The Wall Street Journal and the phone book are a few examples of periodicals.
A magazine is a periodical aimed at the general public and containing news, personal narratives, and opinion. Articles are often written by professional writers with or without expertise in the subject and contain "secondary" discussion of events, usually with little documentation (e.g. footnotes or bibliography). Magazines use vocabulary understandable to most people, and often have lots of eye-catching illustrations. Time, Newsweek, U.S. News & World Report, and Psychology Today are a few examples of magazines.
An academic journal is a scholarly periodical aimed at specialists, academics, and researchers. Articles are generally written by experts in the subject using more technical language. They contain original research, conclusions based on data, footnotes or endnotes, and more often than not include an abstract and bibliography. Many, but not all, academic journals are peer-reviewed. The Journal of Physical Chemistry, The Chaucer Review, The Milbank Quarterly, and Labor History are a few examples of journals.
It's important to understand the differences between journals and magazines. Magazines are not necessarily bad or low-quality (nor are journals necessarily high-quality), but they aren't designed to support most upper-level academic research. They often don't document their sources of information and they generally lack the depth of scholarly journals.
For a more in-depth look at the differences between varying types of periodicals, and how to tell if an academic journal is peer-reviewed, view the files below.