Is it fair that a PhD student should be allowed to review a research paper? I really can't understand why an inexperienced PhD student should be allowed to review a research article written by researchers of considerably more experience and expertise. 

How do journals/editors select referees for submissions?
Yes, it is fair. 


The quality of reviews written by highly experienced researchers is low.
Many manuscripts have obvious flaws that should be spotted by the inexperienced.
Reviewers are (in theory) supervised by the editor.
Novice reviewers are more enthusiastic than experienced reviewers.
If the paper cannot be understood by a PhD student, it will not be useful to very many people.
PhD students may have more specialized expertise because they work on fewer projects, as compared to senior researchers (per @nayrb)


Of course, it depends on the individual PhD student.  Some are better prepared to review than others.