Suppose White has only a king and queen and Black has only a king. First of all, if the kings are connected then the game is drawn as explained in "Connected Kings."
If the kings are separated and White can keep it that way, then forcing mate with the lone queen is easy. For example, consider:
8/8/5k2/8/8/8/8/5K1Q w
White to Move and Win
White wins by forcing the Black king to an edge of the board with his queen (unlike in chess, the White king doesn't participate). White proceeds like this: 1. Qd5 Kg6 2. Qf5+ Kg7 3. Qg6+ Kf8 4. Qf7#.
In this ending, White's first objective is to ensure that the kings stay separated. If it's White's move and the kings are separated by at least two ranks or two files then this is easy. For example, consider:
8/8/8/2k5/8/8/4K3/5Q2 w
White to Move and Win
Here, White cuts off the kings by playing Qf4. Then the mate is easy.
Sometimes it is possible to cut off the kings when there are not two files or two ranks separating them. For example, if the kings diagonally oppose with one square between, sometimes the queen can intercede like this:
8/8/8/8/2k5/8/4K3/1Q6 w
White to Move and Win
Here, White's only winning move is Qd3+. Now the kings are permanently separated and the mate is easy.
When the kings are separated by only one file or one rank (but are not diagonally opposed like above), White can still sometimes win if it is his move. The technique is to run his king toward his queen. For example, consider:
8/8/8/3k4/8/3K4/5Q2/8 w
White to Move and Win
White wins like this: 1. Ke2 (1. Kd2? Ke4 2. Ke1 Kd3 [==]) Ke4 2. Kf1 Kd3 3. Qe2+ and the kings are separated.
Sometimes the journey is longer. Consider:
k7/8/2K5/8/8/8/7Q/8 w
White to Move and Win
1. Kd5 Kb7 2. Ke4 Kc6 3. Kf3 Kd5 4. Kg2 Ke4 5. Kh1 Kf3 6. Qg2+ and the kings are separated.
Or, consider:
7k/8/7K/8/8/8/7Q/8 w
White to Move and Win
1. Kh5 Kg7 2. Kh4 Kf6 3. Kh3 Kf5 4. Kg2 Kg4 5. Kh1 Kf3 6. Qg2+
If the White queen is poorly placed then the endgame is drawn. For example, the following positions are all drawn:
8/8/3k4/8/3K4/8/8/3Q4 w
Drawn
8/Q7/8/8/2k1K3/8/8/8 w
Drawn
7Q/8/8/8/8/2K5/8/k7 w
Drawn
If Black has a draw, it is always easy to spot. For example, consider:
8/8/5k2/8/8/5K2/8/4Q3 b
Black to Move and Draw
Black draws by playing 1. ... Kf5 (not 1. ... Kg5? 2. Ke2 Kf4 3. Kd1 Kf3 4. Qe2+ [1-0]) 2. Ke2 Ke4 3. Kd1+ Kd3 [==].