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Gain is usually defined as an overall boost in your signal without any added tonal coloration. It is usually expressed in decibels such as "Gimme a 3dB boost on that bass." Adding more gain is basically just amplifying the signal so it cuts through the mix or can be heard over a loud drummer.
Overdrive, as it has come to be known as, is a smooth, warm, slightly distorted sound, generally associated with the sound made by cranking up a tube amp. It is fat and dynamic, allowing you to vary your tone just by the way you play. Overdrive pedals can come close to giving you that tube-overdrive sound, probably the most popular being the Tube Screamer.
Distortion can be defined as anything from a fuzz tone to a full-on, notched-out midrange, death metal wall of noise. It is a hard-edged sound with as many variations as there are players. Digital distortion has a more metallic, raspy sound which works well for heavy metal, grunge, or reliving your 80's hair band days. Analog tube distortion gives a good, all-around rock tone, such as the sound of a Marshall stack cranked to 10.
The key to all this is to listen and experiment and let your ears be the final judge. Knowing the differences and how to use them will go a long way to helping you define your own signature tone.