1.O-O. The king travels from e1 to g1, the rook from h1 to f1, and the scoresheet gets one entry for the pair of them. That is the only moment in a legal game of chess when two of your own men change square on the same turn. It is a shortcut, and it exists because moving the king off the e-file and the rook onto it one square at a time would cost three or four turns nobody can spare in the opening.
8 
7 
2 
a 
e h 1 8 7 2 a f g Everywhere else, one piece per turn
The rest of chess is strict about this. A knight cannot hop twice. You cannot advance a pawn and then decide to bring a bishop out with the same turn. You cannot save up moves from a quiet phase and spend two at once when the attack comes, however much you would like to, and you cannot give a move back either: there is no passing your turn when everything you have is bad.
That single-move limit is what makes chess playable. Every threat you make gives your opponent one turn to answer it, so a plan that needs two moves to work can be stopped in between. Half of all chess strategy is counting how many turns each side needs.
Promotion and en passant are one move each
Two other moves get mistaken for double moves.
Promotion looks like two because two pieces are involved. It is not: the pawn reaches the eighth rank and is exchanged for a queen, rook, bishop or knight on that same square, all before you press the clock. Nothing else moves. See what pawn promotion is for the choice of piece, which is not always a queen.
En passant looks like two because a pawn disappears from a square your pawn never touched. Your pawn on e5 answers d7-d5 with exd6, landing on d6 while the pawn on d5 comes off. Only one man has moved. Removing a captured piece is not a move; if it were, every ordinary capture would be a double move too.
At the board, take the king first
Castling has an etiquette that costs points if you ignore it. Because the Laws treat it as a king move, you pick the king up first, move it two squares, then bring the rook round with the same hand. Grab the rook first and a strict arbiter can hold you to a rook move, since a touched rook is a rook you have to move. Junior players get caught by this every season. There is a page on which piece you move first when castling, and the five conditions that have to be met before you are allowed to do it at all are in the castling rules.
Once the mechanics are clear, the useful question is when to do it rather than whether you may. Castle early, castle on the side your opponent is not attacking, and see the castling puzzles where the move itself is the winning idea.