Quick answer
An optional exchange with the player across from you after the Charleston ends. Each of you names a number from zero to three and the lower one is used. No joker may be passed in it.
The Charleston gets all the attention and the courtesy pass gets skipped, which is a shame, because it is the exchange with no cost attached. You can take zero tiles. You can take one. It happens once everybody at the table has already decided what they are building, which makes it the best-informed pass of the hand.
When it happens and with whom
The courtesy pass comes after the Charleston has finished, whether that was after the first round alone or after both, and before the first tile is drawn. It is the last thing that happens before play.
It is across only. You exchange with the player directly opposite, nobody else, which is why it is sometimes called the optional across. The other two players are not part of it and may be doing their own exchange at the same time.
How the number works
Each of you names how many tiles you want to exchange, up to three. The lower number is the one used. If you want three and the player across wants one, you exchange one each.
Zero is a legal answer. If your hand is where you want it, you say zero and nothing moves. That is not rudeness, it is the mechanic working as designed, and it is why this is the only pass in the game where fewer than three tiles can change hands.
Some tables allow more than three. Since the cap varies, settle it before the deal rather than negotiating it mid-hand.
No jokers, still
The rule against passing a joker runs all the way through the courtesy pass. Both of the sources this page draws on state it in the same terms: a joker may not be passed in any Charleston pass, and the courtesy pass is not an exception.
This is worth knowing precisely because the courtesy pass feels like a separate, gentler thing happening after the rules part is over. It is not. If you are holding a joker you do not want, which is a rare and enviable problem, it stays with you until you can use it or discard it in play.
What to actually move
By this point you have been through six or nine passes and you have a direction. The tiles worth exchanging are the leftovers of the direction you gave up on: the lone honour from a hand you abandoned, the third suit you stopped feeding, the odd tile that survived because it was never quite worth passing.
Ask for the smallest number that clears those. Naming three when one would do means giving up two tiles you actually wanted, and the player across is under no obligation to send you anything useful.
Do not break a pair to fill the number. A pair is the part of a hand a joker can never repair, and breaking one to satisfy a count you volunteered is the worst trade available at this point in the hand.
What you get back
You have no control over what arrives, which is the honest limitation of the exchange. The player across is shedding their own dead tiles, and their dead tiles are not chosen with you in mind.
That cuts both ways and it is the reason to keep the number small. A one-for-one exchange is a cheap look at one more tile. A three-for-three is a real reshuffle of a hand you have already shaped.
Put it into practice
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The courtesy pass only helps if you know which tiles are the leftovers of a direction you already abandoned. Mei, our AI coach, reads your hand and says which direction it goes for, what the backup is, and which tiles are doing nothing for either.