The Hobbits before their Wandering Days in the Vales of Anduin had contact with that people and their languages had many in common. For example the Rohirrim had retained the legend of the being known as kûd-dûkan (translated as hol-bytla), a term which became kuduk by the Hobbits, the name they had for themselves.
The latter shows the element tûr also seen in the name Tûrac "People-king".
Despite its relation to Westron, Rohirric was not intelligible to its speakers. Legolas was unable to understand the songs, however he noted that the language is like the land itself: rich and rolling in part, and else hard and stern as the mountains.
As Westron is rendered in the novels with English, Rohirric is always translated through Old English. This is because Tolkien tried to reproduce for English readers its archaic flavour in relationship to the Common Speech.
However the relationships between the two pairs is not identical: Old English is the direct ancestor of modern English, but Rohirric was not the direct ancestor of Westron, since the latter derives from Adûnaic.
Tolkien did not give a known name for the language of the Rohirrim.
Rohirric seems to be an invention of Robert Foster in his Complete Guide to Middle-earth. Perhaps it was modelled on "Rohirrim" and the -ic of "Adûnaic". However the double r has no place here, since it derives from singular Rohir and the plural ending rim.
Tolkien himself used the adjective Rohanese in The Rivers and Beacon-hills of Gondor but it is not clear if he refers to the language or the names.
Lisa Star mentions that Rohirian is found in a manuscript labeled Mq15:10 and also in Peoples of Middle-earth p. 55, which is untrue. It has been also suggested that the manuscript actually says Rohirin but it is just a theory.