The son of Isis and Osiris (or their brother, depending on which version of the story you listen to), Horus has been nicknamed "the avenger" for the role he plays in Egyptian mythology: he avenges his father Osiris by defeating his uncle Set.
The first time Carter sees a manifestation of Horus, here's how he looks: "a young man in leather armor navigated the boat. His eyes were outlined with kohl, and his head was bald except for a braided ponytail. The guy definitely worked out, because his arms were ripped" (26.88). Also, "his eyes were two different colors—one blazing yellow like the sun, the other reflective like the moon" (26.91). Don't forget that Thoth fashioned an eye from the moon to replace one of Horus's eyes, which Set cut out. Ouch.
Losing his father was one of the main motivations Horus had for becoming such a fierce warrior, since "his loss gave me anger. Anger gave me the strength to defeat Set and take the throne for myself" (26.137). According to Horus, Carter must do the same.
That same anger can be dangerous, though. When Sadie hesitates before finishing the spell to banish Set to the Duat, she looks at Carter, seeing the doubt on his face: "Horus would be urging him toward bloody vengeance. I was hesitating. This is what had turned Isis and Horus against each other before" (40.51). Vengeance is good and all, but it seems like Horus has to work extra-hard to make sure it's not misdirected.
With the falcon as his symbol, Horus is known as the war god. So that means he's brave and fearless… but maybe not always the cleverest god in the sky. For instance, when Carter challenges the Set animal, Horus tells him excitedly: "Well done…We will die with honor!" (22.26). Yeah, it turns out that dying—with honor or not—is not on Carter's agenda that day. Or ever, really.
But it just goes to show that gods and humans think differently. Dying is a bummer, but all it means is that your current human host goes bye-bye and you return to the Duat until you find a new suitable host. With that kind of attitude, we can see why Horus is sometimes characterized as arrogant.
In fact, Anubis holds that very same opinion. He tells Sadie: "I don't like Horus… He's loud and arrogant and thinks he's better than me" (28.123). Maybe it's just that their specialties lie in different areas: Horus kills, and Anubis takes care of the funeral arrangements. But Anubis isn't the only one to dislike Horus; Sobek think he's grown too weak to respect.
Carter's also not impressed with how shallow Horus can be. When the kids are on the run from Desjardins in Las Cruces, Horus insistently reminds Carter to wish him a happy birthday, since Horus was one of the five gods born during the Demon Days. Um, bad timing, dude.
At the same time, you kind of have to admire Horus's single-mindedness. Maybe it makes him a little one-dimensional, but his willingness to fight to get stuff done is admirable.