An appeals committee is a panel of experienced players that reviews a director's ruling. Some director rulings involve close judgment calls -- for instance, deciding what bids a player might have made in a complex contested auction if he'd gotten a different explanation of his opponents' system.
Players have the right to appeal any ruling made at their table.
However, this is not an invitation to appeal every ruling you happen not to like. Appeals committees are still bound by the Laws just like the director is. For example, you misclick in the auction, but don't notice until the bidding comes around to you again. Neither the director nor the committee can change the fact that there is a clear rule about when a call can be changed, and you did not notice in time. They can't bend the rules just because they know it was an accident. And they do have the right to assess penalties for pointless appeals that only waste the committee's time.
Most appeals arise when the director has to make a judgment call about how the bidding or play might have gone under different circumstances. He'll try to do his best, of course; but he may not fully understand your biddiing system, or may simply have a different opinion of what was likely to have happened than you do.
Usually your first step will be to raise your concerns with the director at the time of the original ruling. He will be happy to explain the reasoning behind his decision. He may even correct his decision on the spot, if, for example, you point out an obvious line of play leading to a different result that he overlooked when he analyzed the hand.
Do not argue with the director at the table, however. That accomplishes nothing and just frays tempers. If you disagree with the director's decision and don't accept his explanation of why he ruled as he did, he will close the conversation by advising you of your right to appeal. He can probably also give you some idea as to whether he feels it is a close decision that could go either way or not.
Each site has its own procedure for how appeals are lodged and heard. Your first step will normally be to politely inform the director you wish to appeal. He will be able to tell you what happens next. In face-to-face bridge, normally you, your partner, your opponents, and the director will be asked to stay after the game a few minutes to tell the committee what happened. In online bridge, you'll probably be asked to file a written statement making your case, either by email or on a web form. Your opponents and the director will be asked to do the same.
You may ask for an appeal any time between the time of the ruling and 30 minutes after the end of the game. Exactly when your case will be heard (and whether it is handled by email among the committee, by internet chat, phone, etc) varies from place to place is generally out of the director's control. The appeals chairman or club manager should keep you informed.
It is not a personal insult to a director to ask to appeal his ruling. The director won't be a part of the appeals committee, but will help you -- and your opponents -- through the process.
The director and the committee follow the same rules, and have the same general aim: ensuring a fair game for everyone.
The appeals committee has a couple big advantages over a director in trying to reach those goals:
A director usually has to act quickly. He has to keep his game running to a timetable, attend to his other duties of making announcements, moving players for the next round, scoring up the game, and so on, and there may be another table waiting for him to come and help them too. An appeals committee has time to analyze every card that is played in as much detail as it wants; to ask for or look up detailed system notes if needed; to poll as many players as they need to form a picture in their minds of what a "typical" player would do in a suitation.
Because the committee has this greater capacity to explore all the possibilities and get all the facts, they emphasize equity to all involved, whereas the director emphasizes protecting the innocent side from any potential damage. If an appeals committee changes the score the director assigned, this doesn't necessarily mean they feel the director's decision was wrong given the circumstances.
That doesn't mean that "appeals committees are always kinder to the offending side than the director is." It is quite possible that they will agree fully with the director and look unkindly on people who make "sour grapes" appeals. Both directors and appeals committees have the power to assess penalties in addition to adjusting the score on a board. Appeals committees tend to use this power much more frequently than directors.
Appeals committees also operate under one restriction that directors do not. Committees can overrule a director's bridge judgment, but they do not get to decide for themselves how to interpret the Laws. Committee members are, in general, players intimately familiar with the game, but not necessary with the law book. Directors, conversely, are required to know the Laws backward and forward, but may or may not be expert players themselves. If a committee has misgivings about how a director applied the Laws, they may consult with the director or approach the site's Chief Tournament Director.
Appeals hearings have a reputation for being unpleasant affairs. They don't have to be. Yes, appeals tend to happen when people disagree strongly about what should happen. But remember, both the director and the appeals committee are trying their best to be fair to everyone. Furthermore, unlike directors, appeals committee members are usually volunteers, trying to make the game better for their fellow players. Respect what they do. Make their job easier by not abusing the system, by cooperating when you're involved in an appeal, and accepting the outcome without hard feelings.
The basis of the appeals process is laid out in Laws 92 and 93.