Academic presses use peer review for books as well.
The process is fairly simple. You send in your article to the journal, and the editor reads it. For more important journals, the task of the first reading of each article is staffed out either to sub-editors or sometimes graduate students. This first reading ensures that the article meets a minimum bar and allows the editor to figure out who to send it to. The article is then sent to one or more readers who are not directly affiliated with the journal who are in the field of the article's subject. In all cases the author does not know the reviewer's identity and vice versa, although in small fields you can guess.
These reviewers will return the article with one of three 'grades': reject, revise and resubmit, and accept. 'Revise and resubmit' is by far the most common response for many journals. In this case, the author redrafts his or her paper and then it goes back to the third-party reviewers for their approval. This might happen several times. Even if a paper is accepted or rejected, it will also receive comments back from the reviewer which are frequently very helpful.
When a paper is accepted by a journal, it is published at the editor's discretion, often within the year.