1) Softneck - This type of garlic has a neck that stays soft after harvest, so you might see them braided. This is a good type of garlic to grow if you live in a warmer climate. Doesn't love the cold. Strong flavor.
2) Stiffneck - You see this one at the grocery store a lot. There's a ring of cloves around the stem. Does really well in the cold but then again doesn't store as well. Flavor is milder than the soft neck.
3) Great-headed also called elephant garlic - Not too hardy and kind of like a leek. Many say the flavor is more like an onion than garlic as we think of it. Both bulbs and cloves are huuuuuge. I find they just don't have much flavor at all which essentially defeats the purpose of garlic.