In order to sit for the NCLEX you need to apply to the Board of Nursing (BON) in the state where you want to practice. Each state varies slightly, but in general you need to provide:
1. The actual application, signed and notarized by a state notary
2. A letter of completion from your nursing school stating that you have met the requirements of the program
3. In some states, you must provide official transcripts for all your courses.
4. The application fee (varies from about $50 to $200 depending on your state)
So you can only take the NCLEX after you have graduated. You also need to register with a company called Pearson VUE that administers the test. This is a seperate $200 fee no matter what state you live in. After the BON approves your application, they will send Pearson VUE an Authorization to Test (ATT). Pearson VUE will email you this number and you need it to schedule your exam. Then you take the exam and your results are submitted to the BON where you applied. The BON then issues you a license number and a paper license.
The length of this process also varies greatly depending on your state. You can schedule your exam right away, or wait as much time as you would like. If you choose to take it as soon as possible, it depends on what dates are available at your testing center. It might be the next day, it might be two or three weeks away. After you take the test, the results are sent to the BON electronically so it is received by the BON almost immediately. Unofficial "quick results" are available to the test taker through Pearson VUE 48 hours after the test. Again, depending on your state, it can take one day to get a license number, or it can take a month.
I think I've ben exposed to the extremes: In Maine, where I grew up, it take about a week from the time you apply until you get an ATT. Once you take the exam, it's about another week until you get a license number.
In NJ, where I live now, it took my over 8 weeks from the time I applied to even get an ATT. It's been two weeks since I passed boards, and I'm still waiting on a license number. BON says it could be another two weeks.