Groundnuts/peanuts are seeds, from which oil is extracted, or they can be eaten raw, roasted or made into peanut butter.
If you plant a raw one in a pot you can watch the plant grow, and produce a pea-like flower, from which what looks like a stem will grow and extend downwards until it reaches the soil, where it deposits the seed - in effect, the seed plants itself in the ground, hence the name 'groundnut'.
(I've never heard of potato oil.)