ITTF Rules are long and can be found here.
For the rest of us here are some of the important basic rules for a fun game:
How to Serve Legally!
When you serve, the ball must be visible. You should have the ball flat in the palm of your hand. You must toss it up at least six inches, AND not hide the ball with your arm. You must also keep the ball behind the line on your table end. You can't serve with the ball in your hand over the top of the table.
If the serve touches the net, it is called a let serve. Let serves are not scored and are played again.
When serving, the ball must bounce on your side and your opponents side without touching the net. The opponent then returns the ball with it bouncing only on your side of the table and play continues until the ball ends up in the net, or a flower bed. If the ball touched your oppenents side of the table before the flower bed you win the point.
A match is normally the best 3 of 5 games (or 4/7 or 5/9). For each game, the first player to reach 11 points wins that game, however a game must be won by at least a two point margin.
A point is scored after each ball is put into play (not just when the server wins the point as in volleyball).
The edges of the table are part of the legal table surface, but not the sides. So it it hits the corner of the table from above it is in, if it hits it from the side its out.
Flow of the Match
Each player serves two points in a row and then switch server. However, if a score of 10-10 is reached in any game, then each server serves only one point and then the server is switched. After each game, the players switch side of the table. In the final game (ie 5th game), the players switch side again after either player reaches 5 points
Bat or paddle Condition
It has to be in reasonably good condition. For international competition your bat or paddle is going to have to have USATT approval for the rubber!
One rule that you may wish to break is that once the player has served and the point is in progress, you cannot talk! In competition play some players get carried away especially when they think they have won the point and might (gasp!) say something. By the rules, if you talk during the point you lose the point.
Also lot of times new players, and particularly kids, make the mistake of putting their hand on the table while they're playing. You can't do this at any point.