"Ping pong", in any case, is in fact a trade name. Parker Brothers came up with it when they wanted to market the game. Some table tennis players will claim that "ping pong" is now actually a different game entirely; namely, one in which every shot is like a serve in table tennis (the ball bounces once on both sides of the table). Whatever the case, table tennis is, despite its unfortunately common nickname, a real sport. It's even in the Olympics. So there.
Most people whose only experience with table tennis is as a game you play in a bar after drinking about 5 pints, or in your basement on a lazy Sunday when the TV's broken, have no idea of the amount of skill that can be employed in its performance. You need much faster reactions than in tennis, and almost as much stamina.*
Table tennis is the second-most-popular ball sport in the world -- after soccer. This is mostly due to its huge popularity in East Asia, where it was introduced by British Army Officers in the late 1800s. It is the national sport in China, which has traditionally dominated the sport at the international level. (It was not, however, invented there.) It also has a large following in Europe, especially Sweden and Germany.
The basic rules are as follows.
Two players stand at opposite ends of a table divided by a net. The serve must bounce once on the server's side, and once on the receiver's side; thereafter, shots are traded with the ball bouncing only on the receiver's side for each shot made. This continues until a player fails to hit the ball back onto his opponent's side (this may involve missing the ball entirely, or hitting the ball but missing the table with it). Each such failure awards a point to the other player. The first to 21 points wins.
Doubles is also played; it differs from tennis doubles most significantly in that there is strict rotation between partners in making returns.
*One can always spot the table tennis player who played a lot of tennis before being introduced to the table version. She will typically hold the paddle in both hands, and will also mess up the order of play a lot in doubles (see below).