Look at the c-file below. Black has no pawn on it, White has one sitting on c2, and nothing else is in the way. That is a half-open file, and it belongs to Black.
The position is a Sicilian: 1.e4 c5 2.Nf3 d6 3.d4 cxd4 4.Nxd4. One pair of pawns came off in the centre, and each side was left with a file it can use.
8 
7 
2 
a b 
c 
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e f h 1 Three kinds of file
No pawns at all on the file and it is open: both sides can use it, and whoever gets a rook there first usually keeps it. Your own pawn on it and it is closed for you, because your rook is looking at the back of your own pawn. An enemy pawn on it and none of yours, and it is half-open, which is the most interesting case, because now your rook has something to shoot at.
That last one is the plan behind whole openings. In the Dragon above, Black’s rook goes to c8 and the whole queenside attack, sometimes including a rook sacrifice on c3, runs down that file. White’s rook goes to d1 and leans on the pawn on d6.
Getting there first
A rook on an open file is worth roughly nothing until it has somewhere to go. What makes the file real is the seventh rank at the far end of it, or a weak pawn halfway down, or a second rook behind the first.
The order matters. Put one rook on the file, then the other behind it, and the pair is far more than twice as strong. If you have only one rook available and your opponent can contest the file, expect a trade, which is fine when it leaves you the one who gets to use the file next.
Files are named for their letter, so the file with the pawn on c2 is the c-file all the way from c1 to c8; if that naming is new, start with ranks, files and diagonals.
Half-open means a target
Here is the connection most club players miss. A half-open file is only good if the pawn on it cannot be defended by another pawn. A pawn on c2 with a pawn on b2 next to it is annoying to attack, because b2-b3 or the knight on c3 will cover it. A pawn stuck on a half-open file with no pawn neighbour to help is a backward pawn, and it is what rooks eat for a living.
The same idea decides most rook endings: whoever’s rook is behind the open file, and behind the passed pawn, is the one making the threats.