Dynamo works only when the wheel is turning, but I think it's important to have the light on even when stopped at red light or when you have a flat etc. and stop in the middle of the road. (Also, I suppose when the bike speed is very slow, the light would be dimmer?)
Most dynamo lights have built in capacitors that keep the light going dimly for ~5 minutes after you stop.
Brightness increases up to around 15-20 kph but at the slower speeds you don't need as much light.