After someone mails in an application for asylum in the U.S. ("affirmatively," not after having been placed in deportation/removal proceedings,) U.S. Citizenship and Immigration Services will schedule that person for an interview. These interviews take place at either one of the eight U.S. asylum offices or with an asylum officer who is "riding circuit" to a closer location.
How long one waits for these interviews, however, is a huge question as of late. You can check processing times on the Affirmative Asylum Scheduling Bulletin page.
Because of the huge influx of Central American refugees, the Asylum Offices have had to focus on certain priorities over others. The top two priorities currently include:
In many cases--in fact, in every case where you see an asterisk on the Scheduling Bulletin--the office is focusing ONLY on those top two priorities. That means they're so overloaded with applicants that they are unable to make any progress whatsoever on the backlog of other cases. If you are in this backlog, there is no way to predict how long you will wait for your asylum interview.
Regardless of how long you end up waiting, be sure to take steps to protect your status as an asylum applicant. Stay in close contact with your immigration attorney (if any) and send notification of any changes in your address to USCIS. If someday you are scheduled for an interview and fail to show up, your case could be referred to immigration court and you could be ordered deported (removed) even without your being there. Such an order is extremely difficult if not impossible to undo.