Apparently, Rohan makes some fine-looking gentlemen. Tolkien describes the men of Rohan: "Their golden hair was braided on their shoulders; the sun was blazoned upon their green shields, their long corslets were burnished bright, and when they rose taller they seemed than mortal men" (3.6.27). Háma is one of these paragons. He guards the door to Théoden's great hall of Meduseld, and he's a looker.
When Aragorn and Company arrive at the gates of Meduseld, Háma demands that they give up their weapons, including Gandalf's staff. Háma is extremely loyal, and when Aragorn tries to challenge him in Théoden's own hall, Háma retorts, "This is the house of Théoden, not of Aragorn, even were he King of Gondor in the seat of Denethor" (3.6.37). Finally, Aragorn gives up Andúril and Gandalf gives up Glamdring. But Háma lets Gandalf keep his staff, as he is an old man. He adds, "I believe you are friends and folk worthy of honour, who have no evil purpose. You may go in" (3.6.49).
Wormtongue is furious at Háma for allowing Gandalf to bring in his staff, but Théoden continues to rely on Háma's service. When Gandalf suggests that Théoden free Éomer from Wormtongue-induced jail, Théoden sends Háma to do the freeing. Háma is so excited by Éomer's freedom that he does as Éomer asks and brings him a sword right away. Luckily, Éomer wants the sword to offer to Théoden, rather than, say, trying to assassinate his king. Háma seems like a nice guy, but he's not too bright.
Still, we will give Háma's brains credit for one thing: it's he who has the fantastic idea of leaving Éowyn in charge of the people of Rohan while her uncle and brother ride to war. It's about time that we get to see the women of Middle-earth getting some real responsibility in these books. Háma is also loyal to Gandalf, defending Gandalf's apparent desertion of the Eorlingas before they reach Helm's Deep to his fellow guardsmen. He has faith that Gandalf will come as he promises.
Sadly, Háma does not survive the Battle of Helm's Deep. Théoden is sorry to see him go: "Great injury indeed has Saruman done to me and all this land [...] and I will remember it, when we meet" (3.8.35). Théoden casts the first earth on Háma's grave.