The fictional language spoken by the Rohirrim in Tolkien’s works is Rohirric.
Rohirric is the language associated with the people of Rohan. In Tolkien’s presentation, the Rohirrim’s actual language is represented mainly through Old English forms in the English text. This is a translation device: Tolkien treated the Common Speech as if it were being rendered into modern English, while the language of Rohan was represented with an older English style.
That relationship explains familiar names such as Éomer, Éowyn, Théoden, and Meduseld. “Rohirric” is a descriptive name used by Tolkien scholars and reference works for the language, while Tolkien’s texts also discuss the speech of the Rohirrim in contextual terms rather than supplying a large fully documented grammar.
Rohirric should not be confused with Westron, the Common Speech used widely across Middle-earth, or with Sindarin and Quenya, Tolkien’s principal Elvish languages. The linguistic design helps connect Rohan’s culture to an Anglo-Saxon-inspired literary atmosphere.