This, as far as I recall, only relates to the game Blackball. It's not technically two shots. The shot that comes immediately after the foul is a free shot from where the ball is. This can be used to your advantage if you know that you will make the foul and want to go for a double-pool. Otherwise, it's just one great shot.
The term "two-shot" comes from how many balls are required. Before Blackball, all pool games used nine balls. Two balls were removed from the rack and placed next to each other on the table to create a single pocket. Then, one ball was removed from the rack and placed next to the first two balls on the table to create another pocket. This process continued until all nine balls had been played. Since then, eight balls have been used instead.
Two shots on Black refers to making two consecutive balls in a row. If you make the first shot, you must leave yourself enough time to make the second shot or you will lose. However, if you fail to make the second shot, you get another chance since it's now a free shot. This means that you can take as much time as you need to make sure you're going to shoot successfully.
In general, players should never hesitate when shooting.
In most circumstances, you will be given the benefit of the doubt as the shooter, and the shot will be ruled lawful. So, if you sink your opponent's ball after hitting both balls at the same time, it is not considered a foul. Your turn is over (unless you sunk your own ball), and the balls are still in the pockets. The other player gets another turn.
However, if you sink your opponent's ball on purpose or accidentally, this is called a foul. If you commit this infraction during your opponent's turn, the referee will stop the game and call for a replacement ball. You will then have a choice: you can keep shooting with the replaced ball or switch to the original ball that was in the pocket with your opponent's sunk ball. It is up to you how you want to play out the remaining shots. If you switch balls, your opponent gets another turn. Once all the balls have been used, the person who sank their last ball wins the frame.
Now, what happens if you sink your opponent's ball but don't say anything about it afterward? Well, this is known as guilty knowledge. If you have some evidence that you knew your opponent was going to hit their ball, they have the right to claim unfair advantage. In this case, you will be given a warning by the referee and may be disqualified from playing in the event of repeated offenses.
When the following behaviors are included in the specific rules of the game being played, they are considered fouls at pool. If many fouls occur on a single shot, only the most serious is punished. If a foul is not called before the following shot, it is presumed that the foul did not occur.
Making sure the cue ball does not touch any other ball except the one being played - this includes the pocket balls. If an object other than the cue ball touches the playing ball, it is illegal for the player to shoot again until the ball has been cleaned with a brush or towel-moved by another player-off the object's influence.
Calling "foul" too late - the shooter must call "foul" before he takes his next shot. If he fails to do so, then he has given the opponent the chance to block the shot and prevent a foul being committed. This rule applies even if the player believes that he has not touched any other balls. He has still committed a foul because blocking the shot was an option available to the opponent.
Taking more than two shots per turn - in order to give everyone a fair chance at winning, players are limited to taking no more than two shots per turn. If you take more than two shots in one turn, others will be able to take advantage of your extra shots by shooting first.
These are just some examples of fouls at pool.
If the cue ball gets pocketed on a stroke, it is a foul (scratch). The shot constitutes a foul if the cue ball comes into contact with an object ball that has already been pocketed (for example, in a pocket full of object balls). A foul may also be committed if the cue ball misses the object ball completely (the cue ball goes out of bounds). When a foul is called, you should stop playing and wait for the referee to mark the score of the frame on a card. You cannot continue until he has done so.
There are seven common fouls in pool shooting: bank foul, corner foul, double-bank foul, hand pass, illegal motion, miss-shot, and no-ball. Some violations are more serious than others; some result in disqualification of the shooter. All fouls can be penalized by having free shots given to the opponent. The referee has considerable discretion as to how many shots he wishes to award against you, but three free shots usually suffice to eliminate most offenders.
The term "foul" in billiards refers to any violation of the rules.