That's a really good question. It'd be nice if there were a little more transparency in the process.
Here's a sketch of the admissions process at the two schools I've attended (undergrad and graduate).
Caveat lector: these ideas have been pieced together from conversations with professors and administrative staff, and are by no means universal, or even particularly accurate. That said, it seems a decent place to start. Critiques are welcome!
The first phase of the process is generally handled by the administrative staff. They toss out applications that don't meet a certain minimum cutoff determined by the admissions committee -- based on GPA, GRE, PGRE, i.e. the quantifiables. If your application doesn't meet those standards, then "bye." If it's incomplete, then it probably gets put aside or perhaps thrown out. If you pass, your application is forwarded to the admissions committee.
The committee that actually reads your SoP, recommendations, publication lists, etc. gets the applications next. This entire process is likely done by email, with perhaps one or two meetings around the decision deadline. The pool of second round applicants gets split evenly between perhaps 3-6 professors, depending on school size. Each professor is instructed to choose the top X% of the applicants they review for a final cut.
The admissions committee knows they will accept a certain number of people, based on how many spots they have open and how likely they think it will be that people will accept their offer. This depends on the size and prestige of the program, and the other schools that students have stated they're applying to.
That entire process is done multiple times, as applications roll in to the department. Earlier applicants may have a slightly better chance; also, the admissions committee typically has a small pool of fellowships they can award in order to encourage the strongest applicants to choose their school. It's therefore to your advantage (in terms of whether you get in, and how much they'll pay you if you do) to get your application in well before the deadline.
At any point, this process can likely be short circuited by a professor in the department asking for you specifically; this is why networking is, imo, the most important part of the grad school application process.