Any score which the director awards a contestant on a deal, instead of a score actually obtained by bidding and playing out the hand.
There are two different types of adjusted scores: an "assigned adjusted score" is a realistic bridge score for a deal. An "artificial adjusted score" is an arbitrary number of matchpoints or IMPs given because no realistic bridge score can be obtained or estimated on a deal. See below for when the director uses each.
You don't "tell the director what score you want to get on the board" after an irregularity. "I want the score adjusted" should almost never be your opening line when you call the director.
These are the two most common situations where you might call a director to have him investigate the possibility of an adjusted score:
First, the director decides if there was an infraction of the rules. If the opponents have made a legal psychic bid or falsecard, or they have accidentally misbid, no rules have been broken, and the table score will stand, regardless of whether this is a good, bad, or average result.
Second, the director decides if there was damage. Damage only occurs when you receive a score that is worse than the score you would have received had the opponents done nothing wrong. If your opponents do something wrong, but shoot themselves in the foot and reach a disastrously bad contract, you might not even call the director; if you do, he'll make sure the opponents are aware of what they shouldn't have done, but you get to keep your bonus and they are stuck with the bad score they inflicted on themselves. Likewise, if the opponents reach a completely normal result "the wrong way," the score might be allowed to stand.
If the director decides that there was an infraction and you were damaged, he now studies the hand to see what would have happened had the infraction not occurred. To your side, he will assign the most favourable result that was likely had the infraction not occurred, and to their side, the least favourable result that was at all probable had the infraction not occurred.
These will be actual bridge scores that might have been obtained at the table but weren't. An easy case: you blow a trick on defence because your opponent made an unalerted splinter bid but you thought it was a suit, and your lead in that suit got trumped. The director will leave the contract unchanged, but transfer one trick back to your side: -420 instead of -450, or +50 instead of -420, or +100 instead of +50.
Here's a somewhat more complex case, to illustrate the difference between "likely" and "at all probable:"
Suppose you hold Kxx xx AQTxx xxx, and the bidding goes 2 on your right, Pass by you, 4 on your left, Pass Pass Pass. No-one is vulnerable. When dummy comes down with 4-4-1-4 shape, you discover that your opponents were playing Mini-Roman 2, but didn't tell you. Over a natural 2 you had to pass; but over an artificial 2, you would have doubled to show a diamond suit. Maybe this would have made a difference to the final contract, maybe it wouldn't; now is the time to call the director.
The director will listen as you (privately -- don't give UI to your partner!) tell him what you would have done differently had you been correctly informed. Then, most likely, he'll tell you to go ahead and defend 4. He'll ask your partner the same thing.
At the end of the hand, the director will come back and tell you whether or not there will be an adjustment. Depending on what everyone else's cards are, that decision could vary widely. Here are some possibilities:
When an assigned adjusted score is given, the scores given to the two pairs at the table usually balance (as in 5 of the 6 cases above), but on rare occasion they won't (as in the last example.)
An artificial adjusted score is only given when it is impossible to go through the process described above and determine what would have happened. Usually an artifical score is given when a hand becomes completely unplayable - say, a player was seated in the wrong place and saw someone else's cards. Only very rarely is an artificial score given for a hand that was actually played at the table. One of the few exceptions is the use of an illegal system like a psychic control; this can result in the pair using the illegal system having their good score taken away from them.
That depends on why the board was skipped.
If your table was added to the game late, you are simply "starting where everyone else already was," and were only scheduled to play 10 or 11 boards instead of the regular 12. In this case, there is no score at all on the skipped board, and your score for the session is based on the boards you actually play.
If a hand is skipped because of something your side was responsible for that made the board unplayable, you will normally receive a slap on the wrist in the form of an "average-minus", an adjusted score on that board of 40% in a matchpoint game, -3 in an IMP game.
On the other hand, if a hand is skipped because of something someone else did, or because of an accident beyond anyone's control, you are entitled to compensation: an "average-plus", 60% or your average score on the other boards, whichever is higher, at matchpoints; +3 or your average score on the other boards, whichever is higher, at IMPs.
If you are having a good game, having a board skipped with no score and receiving an average-plus have exactly the same effect (your final score is the average of what you got on the other boards you played.)
If your last board of a session is skipped because, say, you were there but your opponents walked out of the game early without permission, you receive average-plus and they receive average-minus.
If the director skips a board at your table because the game is running late, but he doesn't assign the blame for the delay to any one pair, he might do one of several things: