Asklepios, healer-god and son of Apollo, was very likely originally a hero; the worship of heroes in ancient Greece differed from that of deities in numerous ways, one being that a hero was generally a local entity devoted to a particular city or region. By the time Asklepios’ worship began to spread beyond Epidaurus, he had certainly become generally known as a god.
Once he had been granted godhood (sometime before the mid-fifth century BCE), his worship expanded to include much of Greece and beyond, and he enjoyed a great and extensive popularity. A benevolent, accessible deity, Asklepios is part of a family of healer-gods who are less well-known but equally kind and good. (The specifics of the Asklepian family tree do vary somewhat with the source, as do the specifics of which of his family members were honored at different sites.) He was not only a god called upon by individuals in need of healing, he was also the patron deity of physicians.
The god Apollo, father of Asklepios, is also (among other things) a healer, but tends to operate on a larger scale; he is responsible for both the ending and the sending of plagues.
(Koronis, Asklepios’ mother, is–unsurprisingly given that her mythic death is a result of her betrayal of Apollo–not directly associated with his worship or his temple, but she did have her own heroine cult at the city of Titane in Corinth, where at the time of one of Asklepios’ great festivals she received her own offerings, not at her son’s temple but at the temple of Athena.)
Epione, Asklepios’ wife, is concerned with the relief of pain. She played an important role in Epidaurus, the first of his major cult centers.
Epione and her daughters are known primarily because of their inclusion in Asklepios’ cult, rather than their mythology–they all surely received active worship in ancient Greece. (The sons of Asklepios, Makhaon and Podaleirios, are by contrast known primarily from literary references to the Trojan War; they are known as extraordinary physicians but do not generally receive the same sort of worship as his daughters and appear to have been considered to be wholly mortal. It was said of several Asklepian temples that they were founded by descendants of the god.)
Hygeia is probably the best-known of Asklepios’ children; as patron goddess of health itself she is often depicted in art along with her father. In most regions she was his usual associate. Her name survives in the word “hygeine,” which is certainly a requisite for good health.
Iaso‘s concern is with modes of healing–cures and remedies–and with the return to good health of those who had suffered illness or injury.
Panakeia, goddess of the universal remedy, is still known to us in the word panacaea.
Akeso ruled the realm of healing itself, the actual process by which injuries healed and illness was cured.
Aegle‘s name refers to the radiance and glow of good health. (According to some sources “Aegle” was an alternate name for Asklepios’ mother, Koronis; take from that what you will.)
For a bit more info on Asklepios, see here for an article I wrote for Oak Leaves a few years back.