Well, they are living creatures and Elemental type has no special rules for healing. So they just heal as other living creatures do. They do regain hp by natural healing. And healing spells (and harm/inflict spells) work usually on them. They have some rules regarding bringing back from death, though (see MM glossary).
Regarding regeneration, if you are talking about loss of body parts or something, they do not have such things from the beginning. They do not have discernible anatomy. They are just mass of elemental energies. So, they do not, say, lose eye or limb.