First things first, when referring to the mythology of Aries, it is important to note that there are two distinctions to make. Myths about "Aries, the Ram" are different from myths about "Aries, the Greek god" (whose name is more often spelled "Ares").
In Greek mythology Ares is the god of war. It is important to make a distinction between the role that Ares (Greek mythology) plays and the role that his sister Athena (Greek mythology) plays. Athena was also a god (goddess) of war, but unlike Ares, Athena was strategic and disciplined, where Ares was chaotic and destructive.
Having these two Greek gods represent the two different sides of war is very telling as to how the ancient Greeks viewed war. If Athena was Napoleon then Ares was Rambo.
Ares was known for his lust for blood, his chaotic nature, and his thoughtless aggression.
Interestingly enough some of these characteristics ( to a much lesser extent) are attributed to those born under Aries, the Ram, in astrology.
Aries, the Ram, is a constellation of stars visible from earth that appear in what the Greeks thought to be the shape of a ram's head. The mythology of Aries comes from the story of Jason and the Argonauts, whose main quest was to find the golden fleece of Aries the Ram in order to prove himself to be the rightful king of Iolcos in Thessaly.
The ram Aries has nothing to do with the god Aries (Ares), despite the fact that they are both based on Greek mythology and they share the same name. The god Ares was most often shown as a dog or vulture when in animal form, though one of his more famous exploits involved him turning into the shape of a boar to kill the beautiful Adonis (Greek mythology), who was in love with Ares' lover, Aphrodite (Greek mythology). He was not known to take the form of a ram.
Ares myths are essentially the same as the myths of Mars from Roman mythology, who is most often considered to be the same god.