Here is a mate delivered by a pawn, with no promotion involved. White pushes the g-pawn one square and the game is over.
8 
6 A pawn attacks the two squares diagonally in front of it, so a pawn on g7 is hitting f8 and h8. That is a check like any other, and when the checked king cannot move, block or capture, it is mate like any other. The pawn is the smallest piece on the board and it finishes the game.
Look at why that particular one works. The white king on h6 is doing two jobs at once: it covers h7 and it defends the pawn on g7, so the black king cannot simply eat the thing that is checking him. Take the king away and put it on a1, and 1.g7 is answered by Kxg7 with a laugh.
The version you will actually get
Nine times out of ten a pawn mates by not being a pawn any more. It reaches the eighth rank, becomes a queen, and the queen gives mate on arrival.
This is the position every king and pawn endgame is trying to reach, and it is why a single passed pawn is worth so much more in an endgame than the piece values suggest. The pawn is not worth one point there. It is worth nine, one move from now.
Two things to watch when you promote
Check the square, not the habit. Making a queen is right almost always, but a new queen sometimes takes the last flight square away from a king who is not in check, and that is stalemate and half a point gone. In that case promote to a rook instead: he keeps a square, has to move into the corner, and you mate next move.
The other case is a knight promotion that arrives with check, which a queen would not give. It is rare, it is beautiful, and it is the one time you should look twice at what a pawn can promote to before you reach for the queen.
If you want the pawn ending version of all this, pawn endgame puzzles will teach you when a passed pawn actually gets there, and there is more on the pawn’s own powers in can a pawn checkmate.