It's uncommon for the queen to be the last piece standing when everything else has been traded off, but queen endgames do often happen after a pawn endgame, when one or both payers have queened a pawn. When one player has a queen and the other does not, it is usually a very easy win for the side with the queen.
Endgames where both sides have only a queen and pawns can be laborious to win, because progress can often be halted by a long series of checks by the opponent. In general, it is important to centralise your queen and try to control as many squares as possible, to limit the possibilities available for the enemy queen. It is also necessary to keep your king safe from annoying checks.
Passed pawns are of particular value in the queen endgame, as the queen is sufficiently powerful to clear the pawn's path by herself and release any blockades the opponent might set up single handed. Here is an example of this in action:
When the opponent has a passed pawn like this, it can be difficult to stop them from just pushing it all the way through. The main defence in a queen endgame is perpetual check. Instead of making futile attempts to block the passed pawn, the defender should try to deliver an endless series of checks to the enemy king. While there isn't much chance of a checkmate, unless the enemy king is very badly positioned, if he cannot find a safe shelter from the checks and is unable to make progress, he may be forced to abandon the game as a draw.
When you have an advantage such as a passed pawn, but your king is exposed to enemy checks, one way to ease your task is to try and exchange the queens. Your opponent will probably try to avoid queen exchanges in such a situation, but a good way to force the exchange of queens is to use your queen to fork the enemy king and queen.
The mighty queen against the humble pawn may seem a horrid mismatch of power, but there are situations where a well advanced pawn supported by a king can be enough to get a draw. Let's look at a few examples.