The invented language associated with the Rohirrim in Tolkien’s works is Rohirric.
Rohirric is the language of the people of Rohan in Tolkien’s legendarium. Tolkien generally represents it in the text through Old English forms and vocabulary, creating a linguistic relationship between the Rohirrim and the language of the people of Gondor, which is represented through modern English.
This method is an example of Tolkien’s use of linguistic translation within fiction. The apparent English words are not meant to imply that the characters literally speak modern English; they stand in for languages related within the imagined history of Middle-earth.
Rohirric should not be confused with Quenya or Sindarin, the major Elvish languages, or Khuzdul, the secret language of the Dwarves. The term can also be discussed as the language of Rohan rather than as a fully documented language with a large surviving corpus.