When you submit an article to a journal, someone has to determine if it's worth printing. Peer review was developed as a way to screen articles and determine the quality of your article.
At a peer reviewed journal, the editor sends your article out to several reviewers (usually three) who are in the same field, or 'peers'. Generally, your name will be taken off of the article so personalities don't interfer with the process. The reviewers read through your article looking to see if: the topic is unique or novel, if the data or research is sound, and if it's well written. The reviewers can: reject the article; accept it with revisions; accept it as is.
Benefits of peer review are that multiple people decide vs just the editor and the review process weeds out poorly written or researched articles.
Drawbacks of peer review are that it's only as good as the reviewers so poorly written or researched articles have gotten published. Also, peer review was created to look for quality, not fraud.
For more on peer review (I know that someone is interested...), check out Nature Peer Review Debate