The other day, I noticed that a head of garlic I had bought to cook with had started sprouting green as it often does when you've kept it too long. Serious chefs say you shouldn't use it at this point because its, "too bitter."
This is really the garlic saying, "Time to plant me!" Just like potatoes, garlic is started from garlic. In a sense, the individual cloves are the seeds. Plant them green-end-up (root end down), and they'll each grow a new head of garlic.
Garlic is a bulb plant, just like many of the perennials you have in your front yard. In fact, it's a member of the lily family. Like most bulbs, it works best if you plant it in the fall, let it overwinter, and then it'll sprout in the spring and have a full bulb by the end of the summer. According to most of the websites I've found, this is the only way to seriously grow garlic for eating. However, it should produce some small bulbs even if planted in the colder weather at the beginning of spring.
I've started about 6 cloves in individual newspaper pots. I'll transfer them to the garden once we move and maybe we'll have some small heads by fall. Hopefully, I'll remember to plant some more before closing up the garden for the winter so we can have a good harvest next year.