Soldier of the East-Mark under Eomer.
Apparently his second-in-command, as Eomer orders Eothain "Tell the
eored to assemble on the path, and make ready to ride to the
Entwade." Also, he is the only one to openly speak out against
Eomer's lending of the horses, which would certainly not be allowed to
any less than a second-in-command.
Reference: Two Towers: "Book" 3.
(In the Peter Jackson movie, he's a child in Rohan
who rides a horse
his sister to escape Saruman's invasion and give news of it to King