Checkmate is the only way to actually win at chess. You can be up a queen, two rooks, and a moral high ground — it doesn't matter until the enemy king has nowhere to go. Time pressure aside, this is the only goal.
Mate has three ingredients: the king is in check, can't move out of check, and you can't block or capture the attacker. Get all three at once and the game's over. Get two and you've just delivered check — important difference.
Simple mates use a heavy piece (queen or rook) backed up by something — often the king, sometimes another piece. The trick is counting the king's escape squares and cutting them off one by one. Once you can do that on instinct, you can mate from a hundred different positions.
King and rook against a lone king. Bring the rook home to the back rank.
Mini-puzzle
Mate in one. Black's king is trapped in the corner — find the queen move that ends it. (Hint: the long diagonal.)