You are busted in GTA (or you're nicked! in GTA London) when the police has caught you. When you commit to many crimes in GTA, you will get a wanted level, which means the police is trying the stop you. When they do without killing you, you have to go the the police station.
The police confiscates all your weapons when you are busted, and every mission you were doing will be failed. After a brief period, you will respawn in front of a police department. Since GTA III, you'll have to pay some money to bribe the police officers so they release you.