Doubles serving order: who serves next?
Each pair sets its own order at the start of every set. If your pair serves first, one of you serves game one and your partner serves game three; the other pair fill in games two and four. That rotation holds to the end of the set, and receiving order is a separate decision.
How the rotation is set
The pair due to serve the first game of a set decides which of them serves it. Before the second game starts, the other pair decides the same for themselves. From there it is fixed: the partner of the player who served game one serves game three, the partner of the player who served game two serves game four, and so on to the end of the set.
It resets at every set
You are not stuck with the order you picked in the first set. At the start of every new set, both pairs choose again. A pair who lost a set with the wrong player serving at 4-5 can put the other one first and be entirely within the rules.
The same choice is there before a match tiebreak, which replaces a set and is treated like the start of one.
Receiving order is a different decision
This is where most doubles arguments actually start. Serving order and receiving order are set separately, under separate rules. The receiving pair decide which of them takes the first point of the game, the partner takes the second, and that alternation runs to the end of the game and the set.
So you cannot move across to take a better matchup once the set is under way. Your receiving half is your receiving half, and your place in the serving rotation is your place, until the set ends.
In a tiebreak
The rotation inside each team carries on in the order the set had it. Whoever's turn it is serves one point. Then one of the other pair serves the next two, then one of yours serves two, then the other of theirs, and round it goes. Two points each, not one each.
Somebody served out of turn
Points already played stand. If it is spotted during the game, the player who was due to serve takes over immediately. If the game finished before anyone noticed, the order stays as it now is and you carry on with it. A fault served by the wrong partner before the error came to light still counts, so the right server comes in already one fault down, which is the bit that surprises people.
Inside a tiebreak the test is a different one. Rule 27d turns on how many points have been played: an even number and the error is corrected immediately, an odd number and the order stays as altered. Do not reach for the finished-the-game rule there, because a tiebreak is one long game and it will not help you.
Receiving out of turn works on the same principle: the mixed-up order stays for the rest of that game, and the pair go back to their proper order the next time they receive in that set.
In the rule book
ITF Rules of Tennis, Rule 14, Order of Service:
In doubles, the team due to serve in the first game of each set shall decide which player shall serve for that game. Similarly, before the second game starts, their opponents shall decide which player shall serve for that game. The partner of the player who served in the first game shall serve in the third game and the partner of the player who served in the second game shall serve in the fourth game. This rotation shall continue until the end of the set.
Rule 15 does the same job for receiving, and Rule 27 covers what to do when it goes wrong.
Common follow-ups
Can partners swap serving order in the middle of a set?
No. The order is fixed once the set is under way and holds to the end of it. The next set is a fresh choice, and so is a match tiebreak.
Who serves first in the second set?
Whichever pair is due by the normal alternation of games, and then they choose which of the two takes it. Winning the first set does not give you the serve.
Can the same player serve the last game of one set and the first game of the next?
Yes, and this is the one people shout about. The order resets at every set and Rule 14 lets the pair pick whichever of them serves first, including the one who just served. Nothing in the rules stops a player serving two games in a row across a set break.
Where is the server's partner allowed to stand?
Anywhere on their own side of the net, inside or outside the court, and the same goes for the receiver's partner. Rule 26, Case 5 says so outright, with the one condition that they are not creating a hindrance.
Luasc asks each pair once, in the game they first serve, then keeps the rotation itself and offers the choice again at every new set and before a match tiebreak, rather than demanding a tap from a pair who never swap. The board names who is serving and which half they serve from. How it scores doubles.