There are different billiards games that use the 15-ball set. Eight-ball is probably the standard. Fifteen-ball is a game where shooters play for points and scratches make for a dramatic turn of the tables. The game is also sometimes called 61-pool for the number of points that determines the winner.
All 15 object balls (the numbered balls) are racked together, with the 15 ball at the front point of the triangle. Unlike in 8-ball, it doesn't matter how the numbers or solids and stripes are configured in the triangle.
Fifteen-ball is played one-on-one, trading turns after each shot. The shooter must pocket a ball on every shot. Whether you make it or not, turns change after one attempt.
If the cue ball goes in a pocket at any time, it's called a foul or scratch. When this happens, any balls that were pocketed during the scratch are removed and placed back on the table. In pool, this is called being "spotted." Spotted balls are returned to the table on the long side as close to the foot spot they can be without disturbing any other balls.
Each ball is worth its number in points (1 to 15). The shooter collects the points for each ball he pockets, and the first player to reach 61 points is the winner.