Arthegall, whose name suggests the French word for "equal" and "fair"—"égal"—is all about keeping things fair, which makes sense for a guy who's the hero of the book of justice, Book 5. Throughout the book, Arthegall travels around Faerie Land offering his insights into justice as a way to resolve conflicts. While Arthegall seems to represent the judgment and intellectual aspect of justice—figuring out what a just resolution to a given situation might be—his sidekick Talus the robot is the enacting and punishing aspect of justice.
So in many ways, we can see Arthegall as embodying the positive side of justice while Talus has to represents its nitty-gritty, less than pretty side. Spenser may be suggesting that for justice to still exist in our minds as an ideal, we need to be able to separate these two aspects.
As you can find out from the Images, Symbols, and Allegory section, duality and doubling is big theme throughout The Faerie Queene. But for Arthegall in particular, the issue of doubling and mirroring is especially important—which makes sense since this is also an important issue for Britomart, his ladylove. But before we get to her, we need to take note of Arthegall's connection to Arthur, a connection so total they even share the same beginning of their names.
Since Arthegall is destined to marry Britomart and found a long-line of British kings and queens, and since Arthur is a famous king of Britain, we can see Arthur and Arthegall as paralleling or mirroring each other in their association with, if not embodiment of, leadership. Indeed, if anyone other than Arthegall is intervening in conflicts trying to offer resolution, it's going to be Arthur.
But even wackier than Arthegall's mirroring of Arthur is Arthegall's mirroring of (and by) his own fiancée, Britomart. Since Britomart dresses as a male knight—indeed, Arthegall mistakes her for a man when they first meet—she is physically a kind of double for Arthegall. Remember too that Britomart first fell in love with Arthegall when she saw his face reflected in a mirror, so we have both some literal and metaphorical mirrors aplenty here. If extend then this idea of Britomart as looking like a male knight to this moment of love at first sight, there's a strange way in which Britomart falls in love with an image of herself in the mirror.
And there's more! Later in Book 5 poor Arthegall finds himself in a compromising situation when he is taken prisoner by the Amazon queen Radigund and forced to dress like a woman and do women's work: "Then tooke the Amazon this noble knight… And caused him to be disarmed quight,/ Of all the ornaments of knightly name… In stead whereof she made him to be dight/ In womans weedes, this is to manhood shame" (V.v.20). So, Arthegall not only mirrors Britomart when she's knight, but he also kind of mirrors her when she's a woman, since he's been forced to dress like one too.
Arthegall is also connected to the ancient hero Achilles, whose armor he is apparently wearing (ignore the fact that Achilles wouldn't have worn armor anything like the medieval armor worn by Arthegall—not the point.): " And round about yfretted all with gold,/ In which there written was with cyphers old,/ Achilles armes, which Arthegall did win" (III.ii. 25). Now Achilles is pretty much the greatest, most fearsome Greek warrior ever. He's so important that he gets an entire epic written about him, Homer's Iliad. So being compared to Achilles definitely means you are destined for some great things.
But, as anyone who has read the Iliad will remember, Achilles isn't close to being a perfect guy. In fact, sometimes he's kind of jerk. He pouts, he whines, and all in all can act a bit like a child. So Arthegall's connection to Achilles isn't 100% complimentary, it suggests Arthegall has some flaws he needs to work on as well.
Arthegall does tend to get put out of sorts when he loses, which is not a very knightly thing to do. Achilles might also evoke some of Arthegall's gender-role bending qualities, since Achilles' close relationship to his friend Patrocolus has been understood by many readers of Homer to have been homosexual.