There are at least two versions, one from the Golden Age of Comic Books and other from the Silver Age of Comic Books. The common elements of the versions are their equipment and costumes; both wear a special belt that allows them to propel themselves into the air and they wear a harness of large wings that allows them to steer. They also both have a female partner, Hawkgirl (Later Hawkwoman), who serves as a more or less equal partner.
Hawkman has one of the most convoluted continuities of any comic book character, with a variety of new incarnations and rewriting of history to reassign the exploits of one Hawkman to another.
What follows is a simplified accounting of the currently accepted Hawkman chronology (like most comic book characters, his history, powers and even personality can vary immensely in different editorial eras). It should be noted thar large parts of this story are the result of retroactive continuity (or retconning), that is, they were added to the story years or even decades after the character was originally created.
Thousands of years ago, in ancient Egypt, Prince Kufu and his consort Shayera were travelling in the desert in the company of the wizard Nabu and the hero Teth-Adam, when they came across the wreck of a Thanagarian space ship. Nabu would later forge the tools that empower the hero Doctor Fate (who would again later tie back into the Hawkman mythos), and Teth-Adam, empowered by the wizard Shazam, would be the enemy of Captain Marvel, and occassional member and ally of the Justice Society of America.
The Thanagrian vessel was styled with a hawk-like motif, and powered by Nth metal, a mysterious substance with many marvelous attributes. Most obviously, it negated gravity, but its energy also strengthened the souls of Khufu and Shayera, binding them together in their love. When the pair were murdered, their souls lived on and were reincarnated in life after life, always finding each other and often dying violently together, usually at the hands of their original murderer, who was also reincarnated. Finally, in the late 1930s, Khufu was archeologist Carter Hall, and Shayera was socialite Shiera Saunders. Just as Shiera was about to fall to their assassin, she was saved by Carter, who had used the left-over Nth metal from the crash to fashion a pair of wings on a harness, which allowed him to fly. As Hawkman, Carter would become a hero, found and lead the Justice Society, and marry Shiera, who would become Hawkgirl.
As they aged, Carter and Shiera became less active as heroes, and eventually had a son, Hector Hall. Carter briefly served with the Lustice League of America. During this period of inactivity for Carter, a new Hawkman appeared (fortunately for the publishers!) Katar Hol was an intergalactic policeman from the planet Thanagar (the planet which had dispatched the ship whose Nth metal powers Carter and Shiera Hall). This police force used the hawk as their symbol, and also had a winged harness. Katar Hol would eventually be altered in a complex series of events, absorbing his wings into himself, and joining the Justice League. However, he, Carter (who had come somewhat out of retirement) became entangled with the "Hawk God" during the Zero Hours story, and all disappeared.
During this absence, a new Hawkgirl appeared, without an attached Hawkman. This was Kendra Saunders, the great-niece of the original Hawkgirl. It was later revealed however, that Kendra, a troubled youth who had attempted suicide, had actually succeeded, and her soul had passed on. The wandering soul of Shayera had taken up inhabitance of the body, and, possessing all of the biologically stored memories, and, confusingly, believed herself to be Kendra.
Since then, Carter Hall has returned from the dead, reincarnated in a new body that resembles his youthful self, except with black hair instead of blond (perhaps reflecting his connection to the dark-haired Katar Hol). This time, however Hawkman is much darker and grimmer than before, mostly because he also remembers every detail of his many lives.
Currently, Hawkman shares a self-titled monthly comic book series from DC Comics with Hawkgirl, and both appear regularly in the same company's monthly JSA title. Aside from heroics, the non-romance between Carter and Kendra (after millenia of the two wandering souls automatically falling for each other) provides a key continuing plot.
Hawkman is connected to a surprising number of heroes, including Black Adam who once served Khufu, and Doctor Fate (doubly so, as Nabu who created the power items of Fate was once an advisor of Khufu, and because the current incarnation of Doctor Fate is Hector Hall, Hawkman's son, also reincarnated after death).