Unless you are able to show your flight school or instructor a proof of US citizenship on your first day, you will have to go through a TSA background check before you can start flight training.
Who needs a TSA background check?
If you are not eligible for a US passport, then you'll need a TSA background check before you start your initial training (most of the time that's private pilot lessons), before your instrument rating training, before your multiengine training and before a type rating (all separate applications). You do not need the TSA background check to study for or to take a written test, but you do need it before you take your first flight lesson (you can take one discovery flight without the background check, but nothing else).
How do I apply for the TSA background check?
The program is officially called the Alien Flight Student Program, and you'll need to know which flight school you will be using before applying as they need to be selected in your application. Head to the website www.flightschoolcandidates.gov and review their FAQs to prepare for your application. Create a login and fill out an application. Your flight school should be able to guide you through the process.
What should I keep in mind if I need the background check?
First, it does take some time for the background check to be completed, so make sure you apply before you intend to start training (plan for 4 to 6 weeks). You'll need to submit your information, the flight school will need to approve you, then you'll need to submit fingerprints, and wait for the actual background check to take place.
Second, the authorization is only valid for one year, and it is not free ($130 as of 2021 for most applications), so if you cannot complete training within that year, you will need to apply again.
Finally, your authorization is linked to a specific flight school or independent instructor. If you decide to switch, you need to apply (and pay) again. If you want to switch flight instructors but the new one works for the same flight school you are already approved for, then you don't need to apply again. So for foreign nationals, the drawback of working with an independent instructor is that you cannot change easily if it doesn't work out, and there are many reasons why it may not work out with your first instructor. So going through a flight school that has a few instructors on staff will give you a few options without having to go through all the bureaucracy again.
Once you have your authorization, you will be training the exact same way every student is training, so the extra hassle is only before you start your training. This procedure was implemented after the attacks of 9/11/01 to ensure that people wanting to learn how to fly did not have ulterior motives. Even though this can feel like a cumbersome step for foreign nationals, keep in mind that this is done to keep everybody safe in the long run!