Black king on a8, my king on a6, rook on c1, and there is mate in one on the board. 1.Rc8 is it. 1.Rb1, the tidy little waiting move that feels like it cannot possibly hurt, is stalemate. Same rook, same two kings, one whole point of difference.
c The count that takes two seconds
One question, asked before you touch a piece on any move that is not a check: after this, how many legal moves does he have? Not how many squares his king is near. Legal moves. If the answer is zero and you are not mating him, put the piece down and find something else.
When the answer is zero you have two fixes and both are easy. Give him a square, or give check. A check is always safe, because a king in check with no legal move is checkmate, which was the plan.
Count his pawns too. A bare king has only king moves, but a defender with one pawn on h7 has a move as long as h6 is empty, and the day you take h6 away is the day the position quietly turns into a draw. I have thrown away a game like that with a queen and two extra pawns, and I did not spot it until my opponent said the word out loud.
The queen does it in one move
She is the worst offender precisely because she is so good at her job. Drop a queen on g6 with the enemy king on h8 and you have taken g7, g8 and h7 with a single piece. Unless that move is a check, the game is over and you have drawn it. The safe method for that ending, the one that keeps her a knight’s move from the king until he is on the edge, is on king and queen against king, and it is worth an afternoon.
The pawn push that draws
Back that up a couple of moves. My king on g6, the pawn still on g5, his king on g8, Black to move: 1…Kf8 2.Kh7 (step aside and give the pawn the file to itself) Kf7 3.g6+ Kf6 4.g7 Kf7 5.g8=Q+, and the new queen is protected by the king on h7. Same pawn, same two kings, no stalemate anywhere.
The other pawn fix is to promote to something smaller. If a queen would leave him frozen and a rook would not, take the rook: you do not have to promote to a queen.
Once it is on the board
There is no remedy. It is a draw, it is immediate, and no arbiter can help you, which is the whole argument for building the count into your hands rather than your memory. Work through mate-in-two puzzles until the mating pictures come to you before the quiet moves do, and the stalemates stop appearing on their own, because you stop wandering about with a winning position. The rule itself, with both conditions spelled out, is on what stalemate is.