I assume it's points for winning battles at the Battle Maison/Resort, but, 2 things, at the battle Resort, is it still limited to Pokes below lvl 50, and how much bp to you get depending on the battle?(The reason I'm asking is because I need an Assault Vest for my future Rhyperior.)
Every battle that you do at the Battle Maison will give you BP, or Battle Points. Usually after a battle you get 1 or 2 BP, but the more battles you do in a row the more BP you can get per battle. After Battle 20 with the Battle Chatelaine, you can start gaining 3 or 4 BP per match.
Any Pokemon can be used as long as they aren't "big legendaries" no matter what level. If they are below Level 50, they will remain at their level.