Known also as Anglosaxon, a language of the Germanic branch of the Indo-European? family. It is the ancestor of the English language.
Tolkien was a master of Anglosaxon and he uses it thoroughly in his works, notably in the names (and some phrases) used among the Rohirrim.
The reason: Westron, the Common Speech of Middle-earth is translated by English. The Rohirrim however used an archaic relative of Westron. The best way to represent this difference and also the relationship of the two languages, was for Tolkien to represent the latter by Old English.
The names Éomer, Théoden, Holbytla, Riddermark, Woses, Mearas, Ents etc are Old English names, representing respectively those names used by the Rohirrim in their own tongue.
Of course, no way the Rohirrim actually spoke Anglosaxon! Holbytla (archaic name for 'Hobbit') is a translation of the genuine Rohiric kud-dukan (which in Westron is kuduk), Wose represents the actual Rohirric rog, Riddermark represents Lograd, and Theoden's actual name was Turac.