Overdrive is an effect that changes the signal of your guitar to mimic the sound of an old school tube amplifier running loudly and cascading into natural break up. It also works well if you have an existing tube amplifier, helping the amp to work harder and bring out its natural dirt. A good overdrive pedal should also offer the dynamic feel of a tube amplifier, reacting to how you play – harder offering more dirt, and softer retaining clean tones.
The vast majority of overdrives available on the market are lightly tweaked copies of existing circuits, with the most commonly copied being the Tubescreamer, but there are plenty of other designs that have multiple homages to them (Klon Centaur and Marshall Bluesbreaker, to name just two). Overdrive as a circuit tends to be relatively simple, using diodes and sometimes op-amps, though various overdrives will use variations to achieve their specific tone.