To foster is to nurture something. A teacher could foster creativity by providing crayons to every student. You can also foster a child, which means the child lives in your home for a time.
You may have seen photos of a cat taking care of a baby rabbit, or a pig nursing a litter of puppies: those are little foster families. Foster can be a verb (to foster someone or something) or an adjective used to describe a foster family, child, or parent. The word is from the Old English fostrian meaning "to supply with food, nourish, support," which sounds like what those mama hogs were doing. Unlike adopt, foster has a time limit.