Their name is from the same root as the word for horse, ashwa; you can see the same root in the word for the horse-sacrifice, ashwamedha. They are also called the Ashwini Kumaras, the "Ashwin Boys."
The Ashwins received their name because of their skill as horsemen and charioteers. As sons of Surya, the sun-god, they are associated with the sky-chariot that brings the sun at sunrise and leads the sun away at sunset. They are also associated with medicine and healing.
Ashwin is also the name of a month in the Hindu calendar, as well as being the name of a constellation.
The Ashwins are important characters in the Mahabharata because they are the divine fathers of the twin Pandava heroes, Nakula and Sahadeva. Nakula, in particular, has great skill with horses, and during the year of hiding, he disguises himself as a horse-trainer in the employ of King Virata. Because they are the sons of the Ashwins, Nakula and Sahadeva are sometimes called Ashvineya.
You will find two different legends about the Ashwins in this comic book on Reserve in Bizzell Library: Ashwini Kumars: Tales of Devotion from the Vedas.