I currently have a sweet potato that is growing in a one half gallon milk container filled half-way with water.. All you do is rinse off a sweet potato from the store, which they incorrectly call yam. This is to wash off the anti spouting compounds they spray on them....
Next you place the sweet potato in water. About 1/3 to 1/2 should be in the water. The roots will form first. If possible, orient the stem end upward and the tip downward. Soon, little purple buds will begin to shoot out of the top portion. As these buds grow longer, they will need to be hacked off and rooted on their own. These growths are called slips. Up to 50 slips can be cut off of a sweet potato, so you only need to buy a big one for a dollar. You can try just placing the whole sweet potato in the ground and watered like normal, but if the shoots don't root before the sweet potato shrinks completely away, they will die.
If the plant is still alive from last year, just put the whole plant back out, and should probably do fine.