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Eomer is the captain of the Rohirrim, the great horsemen and warriors of the kingdom of Rohan. He's also the nephew of Theoden, King of Rohan, and the brother of Eowyn, herself a great warrior.
Eomer became the close friend of Aragorn, fighting with him at Helms Deep and at the Battle of the Pelennor. When Theoden died, Eomer became the King of Rohan, and through the friendship of him and Aragorn, the kingdoms of Rohan and Gondor were more closely allied than ever.
Why a fanlist for Eomer? He's a loyal friend/nephew/brother/captain, a brave warrior, and a man of integrity -- demonstrated when he continued to fight for Theoden despite the influence of Wormtongue.