I think peer review is a vitally important part of the scientific process. For people who are not familiar, peer review means that you have submitted a scientific article to a journal, and they have sent it to 1 or more experts on the topic who will anonymously review the scientific and intellectual merits of your article. If they see any major flaws, they will recommend that the journal reject your paper. If they consider your experiment scientifically sound, and they believe it will add something important to your field, they will recommend that the journal publish your paper. The final say rests with the editor of the journal, who decides to publish or reject based on all the reviews as well as their own expertise.
It is generally very difficult to get a paper past peer review. If there is a major flaw, you can rest assured the reviewers will pick up on it. If there is a minor flaw, you can still rest assured the reviewers will pick up on it. Often, they will require that you perform additional experiments before they will recommend publication. In my experience, nearly every scientific paper is either rejected outright or sent back for corrections before publication. It is extremely rare that a paper is just accepted. Based on absolutely nothing, I would estimate this happens <5% of the time.
So I think peer review adds a great deal of credibility to scientific articles. I would go so far as to say that if an article is not peer reviewed, I am 100% skeptical about it. If it is peer reviewed, I am much less skeptical of it. I wish I could rely on my own judgment to decide if an experiment was done correctly and the conclusions drawn are reasonable, but unfortunately, it is probably impossible to be an expert on every technique in every field and every analysis that can be done. I am inclined to believe expert peer reviewers when they say they were done correctly.
I do think there are flaws in the peer review process. Sometimes reviewers are unbelievable dicks. Sometimes their own biases lead them to reject papers, or to go a little softer on papers that maybe require more scrutiny. I think either the reviewers should not be anonymous, or the authors should be anonymous as well—unfortunately, there are petty rivalries among scientists that can influence reviewers.
I also think there are inherent problems in the scientific process that would be best solved by asking reviewers to act differently. Most studies that reach publication are statistically underpowered, fail to correct properly for multiple comparisons, and rely on arbitrary thresholds for determining significance. The way the system currently works unnecessarily increases both false positives and false negatives. But that is a whole other rant.
Summary—I put a lot of merit in peer review, but it is not always a perfect system.