In the original, you (Captain Olimar) crash landed on an alien planet. On this planet you discover this UFO shaped object, which Olimar calls Onions (he thinks they look like onions). This Ufo spit out miniture plant shaped objects, which Olimar donned "Pikman" (named after his favorite food I believe).
Anyway, you use the help of these little guys to gather the missing parts of your ship. You have 30 game days, and each day (excluding the first) has a time limit before you must return to your ships, and your Pikmin to their Onions, for those that don't make it back get lost to the nocturnal creatures.
The Pikman can be used for battle ala throw them on an enemy and watch them beat the crap out of it. They can also make small bridges for themselves out of sticks to reach items and such.
Its really innovative, as I have nver seen a single game in existance before this one like it.
Of course, I could say the same for Alundra...