White’s king is on e1, the rook is on h1, and f1 and g1 are empty. The move is O-O: the king goes to g1, the rook hops over it to f1, and the pair of them take up one line of the scoresheet.
8 
7 
2 
a b c 
d 
e h 1 Three O’s instead of two, O-O-O, and you have gone the other way: king from e1 to c1, rook from a1 to d1. Black’s versions use the same files a rank higher, e8 to g8 with the rook to f8, or e8 to c8 with the rook to d8.
The way I keep them straight is to count the squares the rook travels. From h1 to f1 is two, and the symbol has two O’s. From a1 to d1 is three, and the symbol has three. Long castling, long symbol.
Letter O or zero
Both are correct. English-language books and every computer format use the capital letter O, because that is what PGN specifies. The FIDE laws print zeros, 0-0 and 0-0-0, and so do most European publishers.
Nobody at a board will misread either one. What you should not do is invent a third form. oo, castles and K-g1 are not notation, and the last is not even an accurate description, since castling is a king move and a rook move made together.
One move, and it can give check
Castling counts as a single move in the laws, and it is a king move: you touch the king first, and if you deliberately touch king and rook together you must castle on that side when it is legal. That is why it fills one box on the scoresheet with no separate entry for the rook, and which piece you move first is a real rule rather than etiquette.
The move also takes the usual marks. O-O-O+ is common, because the rook lands on d1 or d8 and the d-file is open more often than not. O-O+ needs an open f-file and is rarer. And O-O# exists: a game can genuinely finish with a castling move, which is a lovely way to win one.
If it is the rule itself you came for rather than the symbol, it is on what castling is. And the tactical half of the subject, castling into a mating attack or out of one, is drilled in the castling puzzles.