Busted is a term that has been used throughout GTA games.
GTA1 and GTA: London
In these games you will be busted when police touch you, or you've been pulled out of your car. You will lose all your weapons and multipliers. You can prevent losing weapons and multipliers by obtaining a special jail key card before getting busted.
In GTA2 police will hit you until you lost all your health points, but instead of dying you will be busted. Only policemen will try to arrest you, SWAT Team, FBI and Army will try to kill you. You can get busted easier if your health is low, and vice versa - after being hit by a policeman but not busted, you could be easily wasted. When you get busted the events are similar to GTA1: you lost weapons and multipliers, but when having more than x2 multiplier you will be left with x2. You could find "Get Out Of Jail Free Card" pickup and have your weapons and multi's with you.
In GTA3 era games the scene appears on your screen when a cop/SWAT/FBI points his gun at you. This usually happens when you car gets stuck and a cop is able to run to your car door and open it. Other occasions can be when you fall of a bike or are knocked to the ground, cops will take advantage of this and bust you.
On GTA3 you could avoid being busted by parking you car against a wall and looking away from it by pressing L2/R2 and this would lock the door.
As your wanted level increases and you have the SWAT/FBI/army after you you're more likely to be Wasted as they tend to shoot, and have better weapons, but any law enforcement agency officer can bust you.
The outcome of being Busted is losing all your weapons and $100 as a bribe to be let out. This is similar to what happens when you're wasted.
In GTA IV, Niko gets busted when a cop points his gun at Niko and the player doesn't try to resist. You can also get busted when a cop opens your car door and points a gun at you. In this scenario you can not escape. This is generally only when you are on a low wanted level, about 1-2. At 3 you've probably killed a cop so now they take it personally. At 4+ the FIB and NOOSE get involved, they shoot to kill. As with being wasted you are fined a percentage of your current money.