Queen and King
Since a queen is more powerful than a rook, you would think that checkmating with king and queen would be easier than checkmating with king and rook. Well, yes, that's true, and you can generally checkmate the lone king much faster with king and queen. But it's also trickier.

Since a lone rook and king can force checkmate against a lone king, and a lone queen and king can force checkmate against a lone king, it follows that a lone pawn and king can also force checkmate against a lone king. Provided, that is, that the king and pawn can combine to force a safe promotion to a rook or queen.
Stalemate!
The reason it is trickier to checkmate with a queen than with a rook is because the tremendous power of the queen often gives the weak king a chance to set up a stalemate trap. The best way to show that possibility is to try to checkmate a lone king with a lone queen. Nothing else is allowed on the board for this exercise—not even the strong king. You will soon find that, although the lone queen can push the lone king to the side of the board, there is never a checkmate. There are, however, stalemate opportunities.
Black to move. Stalemate.
With the King's Help
Now add the strong king, and the stalemating opportunities go up! Here are a few, with Black to move. They are all stalemate.
Black to move is stalemate.
Black to move is stalemate.
Black to move is stalemate.
Black to move is stalemate.
Black to move is stalemate.
Black to move is stalemate.
So how can you avoid those nasty stalemates? The best way to make sure you avoid stalemating a lone king is to be aware of the traps and look for them before making your move. Another way of avoiding these traps is to make sure the lone king has a spare square if you don't plan on placing him in check. A third way to avoid these traps is to remember that the queen is a long-range piece, and keep her far away.
The Checkmates
There are more possible checkmates with queen and king versus lone king to go along with the added stalemate possibilities. Here is what they look like:
The White king covers what the queen doesn't cover (b2).
The king and queen are a team.
You've seen this one before.
The plan is very much like the one you used with king and rook, except you can dispense with some parts of it. You still have to use the king as a helper, but you don't always need to keep the opposition.
1. Kd5 Ke3 2. Qf5.
The lone king is forced to give way. 2…. Kd2 3. Kd4 Kc2 4. Qh3 Kb2 5. Kc4 Ka2.
Careful! Do you see that 6. Qb3+ Ka1 7. Kc3 is stalemate? Keep your queen far away! 6. Kb4 Kb2 7. Qh2+ Kc1 8. Kc3 Kd1.
Do you see the checkmate? 9. Qd2 checkmate.
Practice these king and major piece against king checkmates against your computer or a willing partner. Once you know them cold you can go into a position, even quite a complicated position, in which you have an extra pawn, and you already know a good plan of action:
Step 1: Exchange off all the pieces.
Step 2: Push your extra pawn through to promotion.
Step 3: Use your king and new queen (or rook) to checkmate the lone king.

