Visit the journal or publisher's website. Publishers will indicate if a journal is peer-reviewed.
Limit your results to peer-review. Most databases, including the library catalog, have an option to limit results to peer-review. If they don't, try adding "peer-review" as a search term.
Search the library's catalog for the journal. The catalog results should indicate whether a journal is peer-reviewed under the journal's name. If there is no indication, then it is not peer-reviewed.
Ebsco databases indicate whether an article is peer-reviewed (results can be limited to peer-review). If you want to make sure, click Publications within Ebsco databases such as Academic Search Complete and search for the journal. Info will be provided about the journal, including if it is peer-reviewed.
ALSO, just because 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.