Dandelions are seen by many people as a weed. They see it as a pretty weed, but still a weed. The truth is that the entire plant is edible, from the roots to the flower itself.
Dandelion greens can be used in pretty much the same ways as any other greens. They can be eaten raw, boiled, sauteed, whatever.
Some people prefer to blanch them to remove some of the bitterness, but this is not a requirement.
The root can be harvested from late fall to early spring. For medicinal purposes the late fall is better. For food purposes, the early spring is better as the root will be more tender and less bitter. You want to harvest before it starts to blossom. 
The roots should be well scrubbed before cutting and using, and drying is suggested for long term storage.
I have heard of people using the roots for tea, and of it being dried and ground as a flour.
Dandelions are considered weeds, and grow like them. When harvesting, even if you harvest down to the ground, if the root is still intact, it will grow back the next year.
- Taraxacum Wikipedia Entry
- A Blog Entry on Harvesting the Roots
- A Blog Entry on Dandelion Root Tea