i wanted to link you to a nice concise explanation of how potatoes grow, but all of the descriptions seemed to make it sound so intimidating and more complicated than fun...too many rules.
here is what works for me:
1) my neighbor hands me a few little potatoes that have grown eyes (she grew them in her garden and they grew eyes on the shelf in her pantry)
2) i bury them outside in a container (w/ drainage holes in the bottom) under some soil & compost (about 4 inches) and watch the green leafy plants emerge (burying whole potatoes versus pre-cut sets is a common practice in England and...it is simple and i like it)
3) after a few weeks i hill up more soil (i hear you can do this w/ straw versus soil, and that sounds nifty, but i have not tried it yet) around the green leafy plants, leaving a few inches of each plant above the soil
4) the visible parts of the plants eventually turn yellowish brown and slump over
5) i dig* into the soil beneath and around the slumped over plants and find potatoes** of various sizes, including the little potato shown above
what fun! i highly recommend planting potatoes.
* don't wait too long to dig or your tubers may decay
**don't eat any tubers that have been exposed to the sun and turned green, they contain a moderately poisonous substance called solanine