You must participate in the TSA's Flight Training Security Program (FTSP) [formerly known as the Alien Flight Student Program] and undergo a background check if you are a non-US citizen seeking flight training. Note that one introductory / demonstration flight is allowed before participation in the FTSP is required.
Flight Training Security Program
A good source of information to complement the FTSP's Application Guide / FAQ page from AOPA:
AOPA: Aliens and Non-U.S. Citizens Seeking Flight Training
How to complete the application:
Training Event: Category 3 (aircraft weight less than 12,500 pounds)
State: California
Course ID: Not required (leave blank)
Aircraft Type
Initial License or Instrument Rating: Cessna 172, Piper Cherokee (put both of these)
Multi-engine Rating: Beechcraft Duchess
Provider Name: Steve Canby, CFI
Class Name: Initial License, Instrument Rating, or Multi-engine Rating
Estimated Start Date: Use date approximately 1 month from today (FTSP processing typically takes this long)
Estimated End Date: Use date approximately 13 months from today (Training does not typically take this long; FTSP allows 12 months for training)
How to get fingerprinted:
Wait for the fingerprint instructions email to arrive from the TSA. Use the link below and click on "Find an Enrollment Center" to find the nearest fingerprint location.