If you are pursuing a suppressor or machine gun as a civilian you will need to complete a Form 4.  In additional, you will need to send a check for $200 made payable to the BATFE.  Certain sections of the form will be required or omitted depending on which method you have choose to apply.  Methods may vary by state.  All form 4 applications and additional documents required should be sent to:

National Firearms Act Branch - BATFE
P.O. Box 530298,
Atlanta, GA 30353-0298

Once mailed expect to wait 2-3 weeks before you check is cashed.  You may call the ATF branch for a status update in which they will ask for your name (or trust name), serial #, and sometimes dealer's name.  The only information they will provide is if and when your application went pending.  If there is a problem, they will tell you it went to a problem status with a brief reason why. You may ask for the examiners name as well. 

NFA wait times vary depending on the current backlog. Wait times can last anywhere from 3-8 months to be completed.  If you are submitting a form 4, I suggest you added your information to the NFAtracker.