The 15 ball has no special significance except that it's worth more points than anything else. So 61 points (or three-fouling your opponent) wins the rack, and you play to a certain number of racks. Spots are given out as points, not balls, so instead of playing 61-61 (even), you might play 75-46 or 68-53.
What is American Rotation? Here's a good video explanation of American Rotation rules. It is similar to Filipino Rotation (which has more-complex scoring). American Rotation is a 15-ball game developed by Joe Tucker similar to 9-ball and 10-ball, but with slightly different rules.
Eight-ball (8-ball) is a game played with a cue ball and fifteen object balls, 1 through 15. One shooter must pocket balls of the group numbered 1 to 7 (solid colors), while the other shooter must pocket balls from the group 9 through 15 (stripes).
15-ball pool, often recognized in the United States as a pocket billiards game or simply as a pool cues game, traces its origins back to the 19th century. Notably, it carries a set of rules that are somewhat more intricate compared to other standard pool games like the straight pool.
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billiard rules 15 balls