To be honest, Éomer's a lot less interesting than his sister, Éowyn. Oh, we like him, sure. He's a decent guy, and he's good with horses (like all the men of Rohan), so what more could you want? But he's also pretty two-dimensional. We can sum up his character in three words: quick-tempered, tough, and loyal.
(1) Quick-tempered: Éomer almost gets into a punching match with Gimli in the first thirty seconds after they meet. Éomer manages to insult Galadriel, Gimli doesn't take it well (since he has a raging crush on her), and Aragorn has to get between them to keep them from killing each other. Clearly, Éomer is a touchy fellow, and he isn't too fond of strangers riding around Rohan in these uncertain times.
(2) Tough: Éomer is the Third Marshal of the Riddermark (Théoden, King of Rohan, is the First Marshal, and Théoden's deceased son Théodred was the Second). So, he's the next in line to Théoden's throne in Rohan. But he doesn't just sit around the Golden Hall of Meduseld enjoying the royal life. Being Third Marshal is a working position, and Éomer works hard at it. He leads a troop of horsemen who help to patrol the land and keep out Isengard's orcs.
In fact, it is thanks to Éomer's men that Merry and Pippin don't get taken straight to Isengard. When Aragorn, Gimli, and Legolas first meet Éomer, they find him and his men cremating a bunch of orcs they killed. Those orcs? The same ones that kidnapped Merry and Pippin. The Rohirrim ("Horse Lords," the horseback-riding fighters of Rohan) sure don't mess around when it comes to orc-hunting.
(3) Loyal: Éomer stands up against Wormtongue, Théoden's awful advisor, even when it means getting exiled and then thrown in prison. When Gandalf tosses Wormtongue out of Rohan, Éomer immediately rejoins his uncle's army without any sign of annoyance or resentment that Théoden threw Éomer in jail (or at least, let Wormtongue do it without any kind of protest). That's pretty loyal—and pretty forgiving—in our book.
But once Éomer is back at Théoden's side fighting Isengard with the Rohirrim, there really isn't much else for him to do. Mostly, he acts as a foil to Éowyn, who has all of Éomer's pride and none of his official duties as a warrior. Éowyn's frustration with her position as a woman in a masculine warrior-culture is much more complicated than Éomer's quick temper, so she's the one who gets all the character development in that family. Éomer's a solid character, but he mostly fades into the background by the end of The Two Towers.