The Rohirrim were descended from the Éothéod, a race of Men that lived in the vales of the Great River Anduin, but that removed to Calenardhon, after which it was renamed Rohan (Horse-land) after their many horses.
The Rohirrim were tall, blonde, and mostly had blue hair. They prized their horses more than anything, and their entire culture was based around these. They had few cities, but lived in many villages on the plains of Rohan.
They are proud and wilful, but they are true-hearted, generous in thought and deed; bold but not cruel; wise but unlearned, writing no books but singing many songs, after the manner of the children of Men before the Dark Years. — The Lord of the Rings: The Two Towers
The Rohirrim had had contacts with Elves (Middle-earth) in their ancient history, and knew of Eru, but like the Dúnedain they did not worship him in any temples. They did value the Vala Oromë the Hunter highly.
They were ruled by a line of kings descended from Eorl the Young, who had first brought them to Rohan, and in time of war every able men rode to meet the Muster of Rohan.