There are two general types of garlic to choose from: hardneck and softneck. Each has its own strengths, and each is more suited to certain situations than others.
Hardneck Garlic Varieties (Allium sativum var. ophioscorodon)
Hardneck garlic varieties are generally hardier than softneck varieties. They are the best option for northern gardeners. They are also the best option if you want to enjoy garlic scapes in early summer, since hardnecks are the only type that send up a strong central stalk in spring (this is the scape.) Hardneck varieties tend to form fewer cloves per bulb than softneck varieties, but they also are usually a bit larger.
Within the hardneck family, there are nine sub-types of garlics: Purple Stripe, Marbled Purple Stripe, Asiatic, Glazed Purple Stripe, Creole, Middle Eastern, Turban, Rocambole, and Porcelain. The Purple Stripe and Rocambole types are the hardiest, best for gardeners who live in the northeastern U.S. and Canada. Gardeners who live in mild climates will have good luck with Porcelain varieties.
Softneck Garlic Varieties (Allium sativum var. sativum)
Softneck garlic varieties are the best ones to grow if you live in a milder climate. They don't form scapes, and generally form several small cloves per head. They mature quicker than hardneck varieties. Softneck varieties tend to store better than hardneck varieties, so this is the type to grow for long-term storage. Sub-types withing the softneck family include Silverskin and Artichoke varieties.