Although there are guidelines as to what can and cannot be eaten, they are by no means exhaustive, so something unexpectedly problematic may get caught in your braces. If this happens do not panic! First try to remove it yourself, carefully, but if it seems impossible get an appointment with your orthodontist as soon as possible.
What Not To Do
The first thing to keep in mind is a list of things you should not do. First and foremost, do not try to take the stuff out with your tongue, or if you do, do it very very gently. Your tongue can get caught in the appliance, which can damage your tongue and the appliance, and the ensuing wreckage can hurt your gums and your teeth as well.
Do not try and use suction to get the stuff off of your braces either. Your cheeks can get caught in the appliance, and can become lacerated and torn, and the brackets or wires may also become ruined in the process.
Do not try to pick the stuff out with your fingers or fingernails. Your hands can bring bacteria into your mouth, and you can definitely hurt yourself or your braces. Your fingers and fingernails are way too bulky to do an accurate job and will only end up either exasperating the problem.
What To Do
The things that you will need in order to safely remove any material caught in your braces is an intraoral toothbrush, an orthodontic toothbrush, and some floss. The intraoral toothbrush can get between your tooth and your bracket and thus can be used to dislodge or scoop out the offending substances. The v shape of an orthodontic toothbrush helps remove detritus without damaging the bracket. Floss may be helpful, but if you use it do be gentle. Floss can get between the bracket and your tooth, taking the adherent out along with the food detritus. So floss only the parts that need it, and be very careful.
Once you are done getting most of the substance out, rinse thoroughly with mouthwash or warm water. This should take all remaining materials out.