Solid state and tube overdrive pedals
An overdrive is not developed to push your amp into heavy distortion. As said above, main purpose of the overdrive is to push your amp, so the amp wil start to break up. With the gain knob it is possible to add more gain to the signal before it reaches the amp.
Most overdrives are solid-state. They use diodes for clipping. Clipping means that the sound wave is flattened, which results in a distorted sound. More explanation about clipping you will find here.
Some overdrives are based on a tube circuit. They do the same as a solid-state drive. The sound of an tube based overdrive can be a little more compressed.