The Rohirrim were a horse people, settling in the land of Rohan, named after them. The name is Sindarin for People of the Horse-lords (sometimes translated simply as Horse-lords) and was mostly used by outsiders: the name they had for themselves was Eorlingas, after their king Eorl the Young who had first brought them to Rohan.
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 which was granted them in perpetuity by the Ruling Steward of Gondor, Cirion in reward for the assistance that they offered Gondor at a time of great need. At that time Calenardhon was renamed Rohan (Horse-land) after their many horses. By the Rohirrim themselves Rohan was usually called The Mark.
The terms Riders of Rohan and Riders of the Mark are commonly used and refer specifically to their mounted soldiers. The former is a chapter title in The Two Towers. The King's Riders were specifically the Riders who formed the bodyguard of the King.
The Rohirrim were tall, blonde, and mostly had blue eyes. 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 Dúnedain of Gondor believed that the Rohirrim were distantly related to them (having descended from the Atanatári of the First Age) and described them as Middle Men, that being inferior to the Númenóreans in both culture and descent, but superior to the Men of Darkness who had worshipped and served Sauron. However J. R. R. Tolkien calls this a piece of Númenórean fiction meant to satisfy the national pride of the people of Gondor for the surrender of the territory of Calenardhon -- in reality there had been no common ancestry between the people of Rohan and of Gondor.
The Rohirrim had had contacts with Elves in their ancient history, and knew of Eru, but like the Dúnedain they did not worship him in any temples. They seem to have valued the Vala Oromë the Hunter highly, whom they called Bema.
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.
Rohirrim is a collective noun and should be used with the definite article (i.e. the Rohirrim). It should not be used as an adjective.