Root-tip, showing young root-hairs

Root hairs form an important surface over which plants absorb most of their water and nutrients. They are also directly involved in the formation of root nodules in legume plants.

They have a large surface area, relative to the other cells, that help them absorb more water and minerals.

Root hairs are usually an outgrowth of a single epidermal cell that grow out of the roots. In other words, they are one cell thick, and subsequently very fragile. This is why if you rip a plant out of the ground even with its visible roots still attached, the plant will die when replanted. It needs the root hairs' extra surface area.