Éomer is Éowyn's older brother, and became the Marshal of Riddermark by the age of twenty-six, his seat of command in the ancient town of Aldburg (east of Edoras).
As Gríma Wormtongue began to influence King Théoden, Éomer and his sister watched as the kingdom of Rohan began to falter. While Éowyn's actions were limited by her gender and her duties to her ailing uncle, Éomer desperately defended the boundaries of Rohan with his riders.
After the First Battle of the Fords of Isen, in which Théoden's son Théodred was killed, Éomer (who was not present at this battle), became the king's heir, a role he would later fulfill after the Battle of the Pelennor Fields in which Théoden is killed.
Interestingly enough, it is Éomer alone who is responsible for saving his kingdom, had he not helped Aragorn, Legolas and Gimli as they passed through Rohan (odd that he was convinced otherwise, as the men of Rohan had become, by this time, suspicious and paranoid).
After the War of the Ring, Éomer and Aragorn both claimed their rightful thrones and became lifelong friends.