White pushes the rook from a1 to a8, says nothing at all, and the game is over. Black still has three pawns, a king and, in most versions of this position, plenty of time on the clock. None of it matters.
7 
a 
g That is the back rank mate, and it is the first mate almost everyone loses a game to. The rook is not doing anything clever. It is the three pawns, sitting on their starting squares, that do the work. I have lost count of the club games that have ended in this exact shape: the queens come off, both sides think the danger has passed, and one rook walks down to the eighth rank while the other player is busy on the queenside.
Written down, that move is 1.Ra8#. The hash is the mate sign, and it is the last thing on the scoresheet: nobody picks the black king up off the board, because the game ended the moment the rook landed.
The three questions
To prove a check is mate, ask the same three questions you ask to get out of one, and get a no to all three.
Can the king move? Count every square around it, including the ones its own pieces are standing on, because those are exactly the squares that make mates possible.
Can anything capture the checker? Careful with the king here. It can only take the checking piece if that piece is undefended.
Can anything block? A line piece can be blocked, a knight never can, and a piece already standing next to the king cannot either.
Three noes, and it is mate. Two noes and a maybe, and it is just check, and your opponent gets a move.
Do the count on the first diagram. The king on g8 has five neighbours on the board: f8 and h8 are taken by the rook’s rank, f7, g7 and h7 are taken by Black’s own pawns. Nothing black attacks a8. There is no black piece that can reach b8, c8, d8, e8 or f8. Mate.
The mate everybody meets in their first month
8 7 
5 
2 a 
b c e g h 1 Scholar’s mate. Four moves, and it works because f7 is the one square in the starting position that only the king defends. Run the three questions on the final position and every answer is no: the king cannot take a queen guarded by the bishop on c4, e7 is covered by the same queen, d7 has a pawn on it, d8 and f8 have Black’s own queen and bishop on them, and you cannot block a contact check.
It is also a bad opening. Any decent player answers 3.Qh5 with 3…g6, and from there Black develops while the queen gets nudged from square to square. Learn to defend against Scholar’s mate and then stop playing it, in that order.
Move one pawn and it is not mate at all
Here is the same back rank picture with one small change. Black played h7-h6 at some point, for no particular reason, and now the rook check is a check and nothing more.
6 a g Escaping is not the only way a check survives: give Black a rook on the fifth rank instead of the pawn move and 1.Ra8+ is answered by a block on c8 or d8. That h6 escape square has a name, luft, and giving your king one is the cheapest insurance in chess. One quiet h2-h3 beats a lost rook endgame. The back rank mate puzzles are full of positions where the winning move is a queen sacrifice that clears the eighth rank.
Mate almost always takes two pieces
Count the workers in the three positions above. The back rank mate is one rook plus three of Black’s own pawns. Scholar’s mate is a queen plus the bishop on c4 that defends her, plus the black queen and bishop sitting on d8 and f8 in their own king’s way. Something has to give the check, and something else has to take the escape squares away.
That is why a lone queen cannot mate a bare king on an open board, and a lone rook cannot either. Both need their own king to come and help, which is the whole content of the basic mates. A single bishop or a single knight cannot force mate at all, no matter how long you try, and neither can two knights against a bare king.
Chess stops one move early on purpose, so you can never capture a king, even in a hopeless position, and no rule asks you to announce anything when you deliver mate. In a tournament you stop the clocks, sign both scoresheets and go outside.
Start with mate in one, where the only skill needed is counting escape squares. Do a hundred of them and you will start seeing the shape of a mate before you have finished reading the position, which is the point at which the three questions stop being a checklist and become a glance.