Children in my club say it like a magic word: hand already flat on the knight, a mumbled “jadoob”, piece nudged onto its square. That is the wrong order, and it does not work. The word only does its job while your hand is still off the board.
J’adoube is French for I adjust. Article 4.2 of the FIDE Laws lets the player whose turn it is put one or more pieces properly on their squares as long as they say so first, and it offers j’adoube or I adjust as the two ways of saying so. Either is fine. What matters is that the words come out before the fingers go in.
Speak, then touch
Say it, then reach. In that order the touch-move rule never fires, and you can nudge the rook on a1 back onto its square, sit the bishop that is half on c4 and half on b3 where it belongs, or square up your opponent’s pawn on e5 if it is annoying you.
Say it afterwards and you have already touched a piece with the intention of moving it, which is what Art. 4.3 cares about. The move is chosen. Arbiters hear the retrospective j’adoube most weekends and they do not buy it.
It covers any piece on the board. Yours, your opponent’s, one you have to lean across to reach. It also covers several at once, so at the start of a game you can announce it once and put four sloppy pieces straight.
Only on your own clock
The right belongs to the player having the move. On your opponent’s time, hands off, even if their queen is sitting across two squares in a way that makes the position hard to read. Wait for your turn.
That cuts both ways. If your opponent is adjusting pieces while your clock runs, they are outside Art. 4.2, and if they keep doing it they are also distracting you, which Art. 11.5 forbids. Ask once, quietly. Call the arbiter if it happens again, and let the arbiter do the talking rather than starting a conversation over the board.
A knocked-over piece is a different situation and needs no announcement at all. If you send a piece flying, you put it back, and you do it on your own clock rather than your opponent’s. Which is a decent argument for keeping your elbows in.
What it is not
It is not a licence to handle the position while you think. A player who handles four pieces a move is annoying an opponent, and an arbiter who watches it for long enough will start applying penalties.
It is not a way out of a bad move either. Once the piece is on its new square and released, Art. 4.7 has closed the door, and no words help.
I use it twice in a normal game. Once at the start, when the pieces have been set up by somebody in a hurry and half of them sit between squares, and once before castling, so that the rook is genuinely on h1 before I swing it to f1. Get in the habit while you are practising castling puzzles at home and it will be automatic the first time it counts.