There are a few common ways to get into that. One way is to actually submit a research paper. The submitting author is typically asked to build a profile with the particular journal and may occasionally, based on the information collected on the profile, be asked to review a paper. Another way is to be recommended by the submitting author(s) as a possible reviewer. When you submit a paper, you are generally asked to submit a list of potential reviewers who are qualified. You would, of course, need to be perceived as qualified. Another way is to be sufficiently well known in the field that an editorial staff member asks you to review based on your established research interests and reputation.
As a student, it may be difficult to be a reviewer for a national/international journal. Maybe you can make contacts with a locally (state or smaller organization, or perhaps a university level) published journal and express interest.
