White pushes the rook from e1 to e8 and writes Re8#. Nothing else gets written after it except the result.
7 
2 
e 
g The hash is the checkmate mark, and it makes two claims at once: the king is attacked, and there is no legal answer to it. That is the definition of checkmate. The game stops there, nobody captures the king, and no further move is recorded. Underneath goes the score, 1-0 if White gave the mate and 0-1 if Black did.
The same thing with different characters
# is what any modern program produces, because it is what the PGN standard uses. Print is less tidy than that.
The FIDE laws list ++ as an equally good abbreviation for checkmate (Appendix C), which is how many older English books wrote it, and it collides head-on with the ++ that modern annotators use for a double check. Some continental books use a double dagger, ‡. Older editions often just printed the word: “Qxf7 mate”.
Read all four the same way and settle any doubt by looking at the position, because after a real checkmate there is nothing left to play.
In a puzzle book, #2 is an instruction
The hash also turns up in front of a number rather than after a move, and there it means something different. #2 is a stipulation: White to play and force mate in two moves against any defence at all. #3 is mate in three, and so on up.
That distinction matters the moment you start solving. A mate in two is not a good check followed by another one; it is a first move after which every black reply loses on the spot, which is why the mate in two puzzles are harder than they look, and why mate in one is the right place to start.
One practical warning for over the board play: write # only when it really is mate. If the game continues after you have written it, your scoresheet no longer matches what happened, and if the score is ever needed to settle a claim, the arbiter is reading a document that says the game finished three moves ago. When you are not certain, write + and check afterwards. The plus is the safer mark to reach for in a hurry.