The easiest way of determine whether a journal is peer reviewed is by using the library's catalog. Search the library's catalog for the title of the journal, and the results will indicate whether it is peer reviewed. If there is no indication, then it is not peer reviewed.
Most article databases, including the library catalog, have an option to limit results to peer review. If there isn't a limiter option, try adding "peer review" as a search term. Many databases like Academic Search Complete (EBSCO) will also indicate whether an article is peer reviewed.
Visit the journal or publisher's website. Publishers will highlight whether a journal is peer reviewed in the description of the journal.
NOTE: Even thought a journal is deemed peer reviewed does not mean every article in the journal is peer-reviewed. Book reviews, editorials, etc., do not qualify as peer-reviewed.