Main rules

If we want to play TT we should know some rules
They are as follows:

The Racket

Construction
The racket must be black on one side of theblade, and red on the other. If two rubbers are used, that means one rubber must be red and the other rubber must be black. If only one rubber is used (which is legal, but in this case the other side of the bat that has no rubber is not allowed to hit the ball), then it can be red or black, but the other side which has no rubber must be the contrasting color. (Law 2.4.6)
The rubbers must be authorized by the ITTF. You are required to show that your rubbers are authorized by putting your rubber on the racket so that the ITTF logo and the manufacturer's logo or trademark are clearly visible near the edge of the blade. This is normally done so that the logos are just above the handle. (Point 7.1.2 HMO)
Damage to the Racket
You are allowed to have small tears or chips anywhere in the rubber (not just the edges), provided the umpire believes they will not cause a significant change in the way the rubber plays if the ball hits that area. This is at the umpire's discretion, so that means that one umpire may rule that your bat is legal, while another may rule that it is not legal. You can protest against the decision of the umpire (Point 7.3.2 HMO), and in that case the refereewill make a final decision on whether your bat is legal for that competition. (Law 2.4.7.1)
Changing Your Racket During a Match
You are not allowed to change your racket during a match unless it is accidentally damaged so badly you cannot use it. (Law 3.04.02.02, Point 7.3.3 HMO). If you do get permission to change your racket, you must show your opponent and the umpire your new racket. You also should show your opponent your racket at the start of the match, although conventionally this is only done if your opponent asks to look at your bat. If he does ask, you must show it to him. (Law 2.4.8)

The Net

The top of the net, along its whole length, must be 15.25cm above the playing surface. So before training or playing a match, you should quickly check both sides of the net and the middle of the net to make sure that the height is correct (if the umpire has not done this already). Most manufacturers make a device that checks the net height, but a small ruler will do the job just as well. (Law 2.2.3)

A Point

You are not allowed to move the table, touch the net assembly, or put your free hand on the playing surface while the ball is in play. (Laws 2.10.1.8, 2.10.1.9, 2.10.1.10) This means that you can actually jump or sit on the table if you like, provided you don't actually move it. It also means that your free hand can touch the end of the table (which does happen from time to time), just as long as you touch the side and not the top of the table. You can also put your free hand on the table once the ball is no longer in play.
For example, imagine that you have hit a smash past your opponent, who failed to touch the ball, but you are starting to overbalance and fall over. Once the ball has bounced a second time (either on the table, floor, surroundings, or hits your opponent), the ball is no longer in play and you can put your free hand on the playing surface to steady yourself. Alternatively, you could have simply allowed yourself to fall on the table, and provided you did not move the table, or touch the playing surface with your free hand, that would still be perfectly legal.
One thing to watch for is a player who bumps and moves the table while hitting the ball, such as smashing the ball. This can happen quite often and is an automatic loss of the point, and is the reason you should always check that the brakes are on when using a table with rollers, since it makes it harder to accidentally move the table.

Service Rules

Intention of the Service Rules
Nothing seems to generate more arguments and controversy in ping-pong than the service rules. The ITTF are constantly tweaking the service rules in an attempt to give the receiver a better chance of returning the serve. Previously a good server could dominate the game by hiding the contact of the ball, making it nearly impossible for the receiver to read the spin on the ball and make a good return.
Keeping in mind that the intention of the service rules is to give the receiver the ability to see the ball at all times in order to have a fair chance of reading the spin, here is the nutshell version of the service rules. You'll see it's still a pretty big nut though! I've got a more in depth explanation ofhow to serve legally in table tennis, with diagrams and videos, for those of you who want a bit more help.
Visibility of the Ball During the Service
The ball must always be visible to the receiver throughout the serve - it must never be hidden. This makes it illegal to drop your hand below the table when serving, or put any part of your body between the ball and the receiver when serving. If the receiver can't see the ball at any point, it is a fault. This is why the rules tell the server to get his free arm out of the space between the ball and the net. (Law 2.6.5)
Ball Toss
The ball must be thrown upwards without any spin, and nearly vertically (this means within a few degrees of vertical, not the 45 degrees that some players still believe is acceptable). Umpires are more concerned about having no spin on the ball, then they are about having a perfectly open hand. (Law 2.6.2, Point 10.3.1 HMO)
The ball must rise at least 16cm, which is actually not all that high if you check it out on a ruler. One important thing to note is that it must rise at least 16cm from the hand, so lifting the ball up with your hand to your shoulder, throwing it 2cm high and then hitting it on the way down is not OK! (Law 2.6.2, Point 10.3.1 HMO)
Contact with the Ball
The ball must be on the way down when serving - no hitting it on the way up! (Law 2.6.3, Point 10.4.1 HMO)
The ball must always be above the playing surface, and behind the endline during the service. This includes the time of contact. Note that it is not a requirement that the bat must always be visible, so you can hide the bat under the table if you wish. (Law 2.6.4, Point 10.5.2 HMO)
Warnings and Faults
The umpire does not have to warn a player before calling a fault. This is only done where the umpire is doubtful about the legality of the serve. If the umpire is sure the serve is a fault, he is supposed to call a fault straight away. (Law 2.6.6.1, 2.6.6.2, 2.6.6.3) The belief that they are entitled to a warning is a common mistake among players, even some at the elite level who should know better!
Furthermore, the assistant umpire is not allowed to give service warnings at all, so he will either call fault if he believes the serve is illegal, or say nothing if he thinks the serve is legal or doubtful. (Point 10.6.2 HMO)
If you have been warned for a doubtful serve (eg a forehand serve that was possibly hidden), and then you serve a different type of doubtful serve (eg a backhand serve that may not have risen 16cm from your hand), you do not get another warning. The umpire should call a fault straight away. One warning per match is all you get! (Law 2.6.6.2, Point 10.6.1 HMO)

Obstructing the Ball

An obstruction only occurs if a player touches the ball (with his bat, body or anything he is wearing), when the ball is above the playing surface, or travelling towards the playing surface, and has not yet touched his side of the court(Law 2.5.8) It is not an obstruction if the ball has passed over the endline, has passed over the sideline going away from the table, or is moving away from the playing surface. (Point 9.7 HMO) So you can be hit by the ball in front of the endline and still not obstruct the ball, provided the ball is not over the playing surface and it is moving away from the table.

The Toss

When the toss is conducted, the winner of the toss has three choices: (1) to serve ; (2) to receive; or (3) to start at a particular end. Once the winner makes his choice, the loser of the toss has the other choice. (Laws 2.13.1, 2.13.2) That means if the winner chooses to serve or receive, the loser of the toss can choose whichever end he wishes to start at. If the winner chooses to start at a particular end, the loser then can choose to serve or receive.

Change of Ends

If a match goes into the final game (ie the 5th game of a best of five), or the 7th game of a best of seven), then the players are supposed to change ends when the first player reaches 5 points. On occasion, the players and umpires will forget to make the change. In this case, the score stays at whatever it is at the time (eg 8-3), the players swap and play continues. The score is not returned to what it was when the first player reached 5 points.(Laws 2.14.2, 2.14.3)

Hitting the Ball

It is considered legal to hit the ball with your fingers, or with your racket hand below the wrist, or even any part of the bat.(Law 2.5.7) This means that you could quite legally return the ball by
  1. hitting it with the back of your racket hand;
  2. hitting it with the edge of the bat, instead of the rubber;
  3. hitting it with the handle of the bat.
There are a couple of important provisos though:
  1. Your hand is only your racket hand if it is holding the racket, so this means you can't drop your bat and then hit the ball with your hand, because your hand is no longer your racket hand. (Point 9.2 HMO)
  2. You are not allowed to hit the ball twice, so if the ball hits your fingers, and then bounces and hits your bat, this is considered a double hit and you lose the point. (Law 2.10.1.6) If the ball hits your hand and the bat at the same time, then this is not a double hit. It can be hard to tell the difference!
You cannot make a good return by throwing your racket at the ball. You must be carrying the racket when it hits the ball for it to be a legal hit. On the other hand, you are allowed to transfer your racket from one hand to the other and hit the ball, since your other hand becomes the racket hand. (Point 9.3 HMO)
The Free Hand
The free hand is the hand not carrying the racket.(Law 2.5.6) Some players have interpreted this to mean that it is illegal to use both hands to hold the racket. However, there is no provision in the rules that the player must have a free hand at all times, so the use of two hands is perfectly legal, if a little strange! The only exception to this is during the service, where there must be a free hand, since the free hand must be used to hold the ball before serving. (Law 2.6.1) Players with one hand or the inability to use both arms are able to be given special exceptions. (Law 2.6.7) In addition, since it is legal to transfer the racket from one hand to the other (Point 9.3 HMO), at some point both hands would be holding the racket (unless the racket is thrown from one hand to the other), and the player would not have a free hand, so this is another argument for allowing both hands to hold the bat.

Rest Periods

You are allowed a maximum rest period of 1 minute between games. During this rest period you must leave your racket on the table, unless the umpire gives you permission to take it with you. (Law 3.04.02.03, Point 7.3.4 HMO)

Time-Outs

Each player (or team in doubles) is allowed to claim 1 time-out period of up to 1 minute during a match, by making a T-sign with the hands. Play resumes when the player(s) who called the time out are ready, or when 1 minute has gone by, whichever happens first. (Point 13.1.1 HMO)

Towelling

You are allowed to towel off every 6 points during a match, starting from 0-0. You are also allowed to towel off at the change of ends in the last possible game of a match. The idea is to stop towelling from interrupting the flow of play, so you are allowed to towel at other times (such as if the ball has gone out of court and is being retrieved) provided the flow of play is not affected. Most umpires will also allow players with glasses to clean the glasses if sweat gets on the lenses at any time.(Point 13.3.2 HMO)
If sweat gets on your rubber, simply show the rubber to the umpire and you will be permitted to clean the sweat off. In fact, you are not supposed to play with any sweat on the rubber, due to the effect this will have on the ball when hit.

Warm Up Period

Players have a 2 minute practice period on the table before starting a match. You can start after less than 2 minutes if both players agree, but you cannot warm up for longer. (Point 13.2.2 HMO)

Clothing

You are not permitted to wear a tracksuit during a match unless given permission to do so by the referee(Point 8.5.1 HMO) Wearing bike shorts underneath your normal shorts is generally allowed, but it is recommended that they should be the same colour as the normal shorts. Again, this is still at the discretion of the referee. (Point 8.4.6 HMO)

Conclusion

These are the main rules that beginners should know, and generally find most confusing. But remember there are plenty more rules that I haven't mentioned, so make sure that you have a good read through the Laws of Table Tennis to make sure you are familiar with them all. I'd recommend having a quick look through the ITTF Handbook for Match Officials too when you can. If there are other questions you need to ask, feel free to email me and I'll help to explain what you need to know.