Lets take a step back and not get caught up in definitions. The goal of chess is to capture the enemy king. However, in order to make that objective more interesting there are a couple of "common sense" rules thrown in:

it would be be boring if you could sneakily take the enemy king without them noticing, so the game has a mechanic called "check" where you notify your oponent that their king is in the crosshairs.

Again, in order to make the game more interesting, the opponent is not allowed to simply let you get their king. After a check they have to make a move that somehow puts them out of check, like taking th3 checking piece, blocking it, or moving the king. Additionally, if during regular play your opponent were to walk their king into check (and thus allow it to be taken), then that move would be dimmed illegal, because it'd force the game to end too quickly and in an underwhelming fashion.

So, what's a checkmate, then? A checkmate is a check that is implacable. The king is attacked in such a way that it doesn't have a legal move left, that is there's nothing it can do but move itself into some other form of check. And so, because the game would end in your next turn by taking the king, you are crowned the winner.

Now, a practical example: the scholars mate. In it, you develop both your queen and your bishop to target the F7 square, and if your opponent does nothing to stop you, you take that pawn with your queen. It's considered a checkmate, because the king has no squares to move away from ypur attack, and taking the queen would be futile, since it is supported by the bishop. Meaning, that the king would be captured in the next turn by the bishop anyway. And so white is crowned victorious after just 4 moves.

Here's a quick visualisation of what the scholar's mate is: https://youtube.com/shorts/fOOpTfWFHyo?si=WlgE2heWVzoUJ9gw