Who breaks
The lag winner chooses who breaks the first rack; after that the break alternates.
Ten object balls, played in numerical order, and every shot is called. Win the rack by legally pocketing the 10 on a called shot when it's the last object ball on the table.
The short version
The lag winner chooses who breaks the first rack; after that the break alternates.
The ten balls are racked tightly in a triangle with the 1 at the front on the foot spot and the 10 in the middle; the others go in without a deliberate pattern.
You break with ball in hand behind the head string. If nothing is pocketed, at least four object balls must reach a rail or the break is a foul.
As in 9-ball: on the shot after a break without a foul, the shooter may declare a push out. Hitting the lowest ball first and reaching a rail are not required; if there's no foul, the opponent decides who shoots next.
Every shot except the break is called: name the ball and the pocket. There is no “safety” call in 10-ball.
If you pocket a ball but don't make your called shot, and don't foul, your opponent decides who shoots next.
Make your called shot and you keep shooting; any other balls that drop on that shot stay down (except the 10). Miss or foul and play passes over. You win the rack by legally pocketing the 10 on a called shot when it is the only object ball left.
The 10 is spotted whenever it's knocked off the table or pocketed on anything but the final, winning shot. No other ball is ever spotted.
After a foul your opponent gets ball in hand anywhere on the table. The fouls are the same as in 9-ball: scratch, wrong ball first (lowest-numbered ball first), no rail after contact, no foot on the floor, ball off the table (only the 10 is spotted), touched ball, double hit, push shot, shooting while balls move, bad cue ball placement, playing out of turn, slow play and the rack-template foul.
Three fouls in a row lose the rack. For unsportsmanlike conduct the referee chooses a fitting penalty.
The player who broke this rack breaks again.