Character Analysis
Ajax, a commander in the Greek army, is the play's official meathead. Literally. Thersites calls him "beef witted" (2.1.12). (We don't advise going up to your school's star linebacker and taunting him that, BTW.)
In other words, Ajax is ferocious on the battlefield, but he's also not very bright. Of course you'll be wanting an example, so here it is: he has no absolutely no idea he's being used by the Greek leaders when he's chosen to face Hector in man-to-man combat. (Remember, Ajax is picked because the commanders want to make Achilles jealous, not because Ajax is the best warrior.) That's why people are always ragging on him. Check out what Alexander has to say:
This man, lady, hath robbed many beasts of their
particular additions; he is as valiant as the lion,
churlish as the bear, slow as the elephant: a man
into whom nature hath so crowded humours that his
valour is crushed into folly, [...]
he is melancholy without cause, and merry against the hair: (1.2.19-27)
In other words, Ajax is "valiant" (bold and courageous) all right, but the guy is also a moody hot-head, which tends to turn all that "valor" and bravery into "folly." By the way, Ajax is always being compared to some kind of animal. Notice how Alexander says he's "valiant as the lion" but "slow as the elephant"? This suggests that Ajax is no better than, well, a brutish animal. This idea speaks to a larger issue in the play: warfare is just a bunch of wild animals going at it (5.7.9-12).
Another thing about Ajax is that he's got both Greek and Trojan blood. In fact, he's related to the Greek warrior Hector. That doesn't stop him from facing Hector in combat, but it does stop them from killing each other. Whenever Ajax's mixed heritage comes up in the play, it seems like Shakespeare is reminding us that the Greeks and Trojans have a lot more in common than they think.