It definitely depends on the program and the school. Yale English, Stanford psychology, MIT biology, U Chicago economics, etc. all receive several hundred applications for 5-10 slots...much, much more competitive than undergrad.
For MA/MS degrees, it also depends. Many graduate programs refuse to admit MS-only applicants and merely grant it to you along the way to your PhD. It's also true that many masters programs are not terribly competitive (geology, for example). However, many are, especially at top schools.
If you are accepted into a PhD program, you are usually given at least 4-5 years of funding. The package usually includes a tuition/fees waiver, a stipend ($15-20k per year), and possibly health insurance or other benefits. In return, most grad students serve as teaching assistants or research assistants.