Peer review is a process whereby an article, before being accepted for publication in a scientific journal, is reviewed by several scientists (peers) knowledgeable in the field to which the article pertains. It is considered a cornerstone of modern scientific research.
When a scientist submits a paper to a scientific journal for publication, the editor of the journal selects several reviewers and sends them the paper to review. The reviewers send the editor their comments and questions about the paper. The editor then decides whether to reject the paper, publish it as-is, or send it back to the author with a list of problems to correct before the paper can be accepted.
In many cases, and especially with reputable journals, the editor removes the name of the author before sending the article out to the reviewers. The purpose is to help ensure that the reviewers judge the paper on its own merits, and not on the reputation of its author.
Strengths of Peer Review
Peer review helps ensure that grossly inaccurate or poorly-reasoned papers are not published in research journals. Thus, the reader of a peer-reviewed journal is assured that the articles are at least good enough to be considered worthy of publication by several experts in the field.
Contrariwise, if an article has not been peer-reviewed, this can arouse suspicion: if the paper is any good, why hasn't it been peer-reviewed?
Weaknesses of Peer Review
Since an article must be read, understood, and reviewed by the reviewers (who are usually working scientists, and thus have full-time jobs already), peer review can delay publication of a paper by months or even years, especially if the editor sends it back for correction. This can delay advances by others in the field.
Peer review is only as good as the reviewers. These may be unfamiliar with the subject treated in the paper (e.g., if a paper on the social impact of economic policies is submitted to a social science journal, the reviewers may not recognize errors in the parts of the paper that talk about economics).
Scientists are only human, and a paper that strays too far from the conventional wisdom in a field may be dismissed by experts, especially if the evidence supporting the article's ideas is weak.