This 3 minute video describes three ways to answer this question:

1. Use the features of  the database in which you find the article. This varies from database to database and you will have to experiment.
2. Ulrich's Periodical Directory will tell you if the journal is peer reviewed. (Since this video was made, the route to Ulrich's has changed but the method remains the same.)
3. Read the editorial material in an issue of a journal or on a journal's web page, usually under a heading such as 'instructions to authors' or similar.

Not all journals are peer reviewed. If the three techniques in the box above yield no results, try a different journal title.

It is also possible that some contents of a peer reviewed journal will not have been peer reviewed. For example, editorials may not be peer reviewed but research articles generally will be. This varies from journal to journal; look for "about the journal" or "instructions to authors"  or "journal homepage" for guidance on this.

Examples

• see the statement on peer review at Science ("...only some of the papers are reviewed in depth....")
• see the statement on peer review at Nature ("...the following types of contribtion...are peer reviewed...other contributed articles are not usually peer reviewed...."). Note especially the 2nd and 3rd paragraphs.