Cutthroat
It's an easy 3 player game. Each person gets one group of balls: low (1-5), mid (6-10), or high (11-15).
The way friends and I play, we all lag for the break and closest person to bottom rail breaks. The first person that makes a ball (could be the breaker or one of the other players), chooses the group he/she wants. The second person to make a ball (if they get the chance), chooses the group he/she wants. The third person gets the remaining group by default.
The object of the game is to be the only person with at least one ball from your group on the table. For example, if you break, make a ball (say the 6-ball), and choose the low balls, you should try to pocket your opponents' balls (7-15 now since the 6 went on the break). If you mess up position where you can't shoot at an opponent's ball but can make a ball from your group, you can pocket your own ball, play position for an opponent's ball, and continue shooting. You get to keep shooting as long as you make a ball.
If you scratch or commit a foul, each of your opponents gets to spot a ball from his/her group that was pocketed earlier.
It's a very easy game, but you can do things to make it more difficult, such as:
1. Make it call ball and pocket.
2. Give weight (for example, you only get 3 balls [1-3 or 7-9 or 13-15] and each of you opponents gets 6 balls... or you only have 1 ball and each opponent has 7... etc.)
Since you have 4 people, one person will have to sit out, but you can make it so that the first person out has to sit out the next game.
Also, a person that goes out (has none of his/her balls on the table) can get back in the game if one of the other players scratch because he/she will be able to spot one of his/her balls.
On a foul, you can play where the incoming player gets ball-in-hand or has to shoot from the kitchen.