Look at any move ever written down: e4, Nf3, Qxh7. Every one of them ends in a letter and a number, and those two things are the whole coordinate system.
The letter is the file, one of the eight columns running up and down the board between the players, a on White’s left through to h on White’s right. The number is the rank, one of the eight rows running side to side, with rank 1 the row White’s rooks start on and rank 8 the row Black’s start on. Letter first, number second, every time. e4. Never 4e.
That ordering is the one thing to fix in your head, because it is what makes chess notation readable at a glance: the letter tells you how far across, the number how far up.
Diagonals, and the two long ones
A diagonal is a slanting line, and here is the thing that surprises people: every square on it is the same colour. It has to be, because a diagonal step changes file and rank together.
g Those two are the long diagonals, eight squares each, and they are the reason a bishop dropped on g2 or b2 in a fianchetto is such an awkward piece to face. Most diagonals are shorter. The a2 to g8 diagonal has seven squares on it, and that one is worth knowing by name, because it points straight at f7, the square that decides half of all beginner games.
Why players go on about open files
A file with no pawns on it at all is open, and it belongs to whoever puts a rook on it first.
7 
2 
e g Count the squares that rook attacks: seven up the file, plus whatever it sees along the first rank. Now put a white pawn of your own on e3 and that rook sees exactly one square up the file. That is the whole idea behind an open file, and it is why in a real game you spend moves swapping a pawn off just to get one.
Rank 7 is the other line with a reputation. Black’s pawns start there, so a white rook that lands on the seventh attacks them where they stand and cuts the black king off on rank 8 at the same time. Two rooks doing it together is close to winning on the spot, which is why the ending pages spend so long on rook endgames.
The habit to build is small and costs nothing: whenever you look at a position, name the file your rook is on out loud. If a pawn of yours is sitting on it, ask what it would take to move that pawn somewhere else.