@talljasperman, do you want to know how to peer review articles OR to write them?
Those who peer review articles would generally have a strong research and publication history in that area or with that journal. They will be recognised as having specialist knowledge in their field. You can find out more about peer reviewing and the process though. This article talks about the peer reviewing process in Science
If you do a search for peer reviewing in the sciences (or be more specific) I think you will find more about this process in other fields.
As to writing papers, check the submission guidelines for journals/magazines you are particularly interested in or like to read yourself. They usually give you fairly specific guidelines on what they want. Also, analyse the way the papers they have published are put together.
There are also a lot of books available (probably through your own library) about writing journal articles. Once you have something written though, try to get it unofficially peer reviewed by someone you trust who shares an interest and knowledge in the field before you submit. You want someone to critique your work who understands your field but also perhaps get someone who is interested but is less of a specialist to read it for meaning.
Good luck. Hope this helps a little.