If you are fortunate to be a Queen ahead at the end of the game, congratulations! It means that you should be able to checkmate the enemy King.
There are many books that can show you how to checkmate, but I prefer this method.
It’s called the dance of the Queen (or the Chess Tango).
First of all, please note that the enemy King can only mated on the edge or corner of the board. We will let Black play first as this will show the method clearer than if White was to move.
Black moves away from of the side of the board and plays 1…Kd5.
Since Black did not move backwards, White just moves his King closer to the action; 2.Kg2. Black moves again with 2…Kc6, or one diagonal square backwards. White’s Queen moves in the same direction as the Black King, in other words she dances with him; 3.Qd4. We are now doing the tango.
Play might then continue as 3…Kc7 (one square backwards) 4.Qd5 (one square forwards) 4…Kb6 5.Kf3 Ka6 (one square to the side) 6.Qc5 (one square to the side) 6…Kb7 7.Qd6 Ka8.
At this point, The King is already on the Back rank, White’s bes plan is to let him just move along the back rank. So 8.Qd7 (watch out, 8.Qc7 is stalemate!) Kb8 9.Ke4 (White just moves his King closer) 9…Ka8 10.Kd5 Kb8 11.Kc6 Ka8 (what else?) 12.Qb7mate.
This is only slightly harder to win. The enemy King again must be driven to the side or corner of the board. And both the King and Rook must do the tango.
First of all, if it is White to move, he makes the dance floor smaller with 1.Rg5. Notice how the Black King simply cannot run away to b5. He is forced to dance.
So Black plays 1…Kb6. If Black King moves to c7 (one square backwards), the White Rook dances to g6 (again, one square forward).
Since the King has move to the side, White dance with his King with 2.Kc4. And so it goes on with 2…Ka6 3.Kb4 Kb6. And now that both Kings are so close enough to see each other (Kings have very bad eyesight), the Rook comes over to check up on them. Yes, its a bad pun, but 4.Rg6+ is the only way to make progress.
The rest of the game could continue as such: 4…Kc7 5.Kc5 (I want to keep my eye on you!) Kd7 6.Kd5 (to continue the dance!) Ke7 7.Ke5 Kf7 8.Ra6 (got to save the Rook!) Kg7 9.Kf5 Kh7 10.Kg5 Kg7 (he ran out of room!) 11.Ra7+ (needs to check up on the monarchs) Kg8 12.Kf6 Kh8 13.Kg6 Kg8 14.Ra8mate