Absolutely anyone can have work published. I published my first article as a PhD student, so I didn't have a PhD at the time.
Once submitted, the work will go to an editor who passes it on to a referee (or referees) that s/he thinks are suitable to review the article. This process is called "peer review". The referee (or referees) will decide if they feel the article is of sufficient quality and originality to be published. The referee will then pass his suggestion on to the editor who takes the final decision.
Of course, the required standard varies between journals.