Point at the bottom left corner of the board with White sitting behind it. That is a1. One square to the right is b1, one square up from a1 is a2. Every square on the board is named the same way, the letter of its column first and then the number of its row, in lower case, with nothing in between.
8 
7 
2 
a 
b 
c 
d 
e f g h 1 The columns are files, lettered a to h from White’s left to right. The rows are ranks, numbered 1 to 8 starting from White’s edge, so White’s pieces begin on ranks 1 and 2 and Black’s on ranks 7 and 8 (FIDE Laws, Appendix C.4 and C.5). Sixty-four squares, sixty-four names, no repeats anywhere.
Both players use White’s map
This is the part that feels wrong for about a week. The names are fixed to the board and not to your seat. Play Black and rank 8 is the one nearest you, rank 1 is across the table, your king still starts on e8, and your scoresheet still says e8.
There is no Black version of the coordinates. The system that did give every square two names, one for each player, was descriptive notation, and the confusion it caused is much of the reason it died.
Lower case for squares, capitals for pieces
b4 is a pawn going to b4. Bb4 is a bishop going to the same square. The only thing separating them is the capital letter, so a square is always written in lower case and a piece letter always in upper case. Write B4 on a scoresheet and you have written nothing at all.
Once you notice the pattern, the names tell you the colour of a square too. a1 is dark and h1 is light, which is why the light square belongs in your right-hand corner when the board goes down. The a1 to h8 diagonal is dark all the way, the h1 to a8 diagonal is light all the way, and a bishop that begins the game on c1 will stand on a dark square for the whole of it.
The squares worth knowing cold
e4, d4, e5 and d5 are the centre, and a great deal of opening theory is an argument about who gets to put a pawn on them.
f7 and f2 are the soft spots: at the start of a game each is defended by nothing but its own king, which is why so many beginner games finish there and why the f2 and f7 attacking patterns repay an hour of anyone’s time.
The first and eighth ranks are where kings get mated behind their own pawns, so learn to see a1 to h1 as a line rather than eight separate squares.
A drill I use with children, and with adults who will admit they need it: call out a square, they point at it, then say whether it is light or dark. Two minutes a day for a week and the counting along the edge stops. After that, reading notation is a twenty-minute job rather than a chore.