It is a simple and complicated process..I know that makes no sense but it is true.
Basically, it is just a matter of paying the tax stamp and passing the background check.
First you buy the silencer. Then you fill out all the ATF paperwork for the transfer from your dealer to you, get your local CLEO to sign off on the paperwork (Form 4), get a passport photo, get fingerprinted, and submit the paperwork and fingerprint cards along with a $200 check to the BATFE. Then you wait....and wait...and wait...and wait some more.
Assuming you are not a criminal, and your check doesnt bounce, and your paperwork is filled out correctly, it will take anywhere from 2 to 5 months to get approved. The BATFE will then mail your approved Form 4 back to your dealer with a cool looking $200 tax stamp affixed.
At that point you can pick up your silencer (or machingun, or whatever NFA item you purchased - the process is the same for all NFA).
Understand that this is a very simplified explanation and does not address the differences in using a trust, various state laws, etc. Depending on your state, county, and CLEO this process can be super easy (Tennessee, Mississippi, Arizona, Nevada) or difficult up to impossible (Hawaii, Illinois, Massachutses, New York, New Jersey).
Check out
www.silencertalk.com forums for much more info. Hope that helps somewhat.