Old English (also called Anglo-Saxon, though J.R.R. Tolkien apparently avoided the term) was the ancestor language of modern English. It was also the mother language of characters Ælfwine and Eriol.
Anglo-Saxon was typically written in a runic script before the introduction of the Latin alphabet.
In earlier notes concerning the The Book of Lost Tales Part Two, Tolkien commented that Old English was the only language the Elves of Eressëa could talk to Men, and that was how they talked to Ælfwine. The Elves learned Old English while living with Men in Luthany.
Tolkien wrote in Old English several texts of his legendarium, which he (fictionally) attributed to Ælfwine such as the Earliest Annals of Valinor; they were published more recently in The Shaping of Middle-earth and are commented and edited by Christopher Tolkien. These texts have been criticized because Christopher Tolkien did not provide a translation and they remain understandable only by Anglo-Saxon students.
In those works, Eriol gave several Old English names to several elements of the legendarium; in some of them, Tolkien attempted to imitate the Elvish sound and form, but with a new meaning. For example Angband is rendered as Engbend "cruel-bond", Balrog as Bealuwearg "baleful-monster" or Silmaril as Sigelmaerels (actually referring to the Nauglamír). At least one name, Mickleburg, survived in the published Silmarillion.
For Lord of the Rings, Tolkien used several Anglo-Saxon names and words, which represented Rohirric, the language of the Rohirrim. This simulated the archaic sense the Hobbits felt with the Rohirrim; Hobbits spoke Hobbitish Westron, represented in the book by English.
However the relation of Old English and English is not the same as with Rohirric and Westron, since the latter descends from Adûnaic, not Rohirric.
 See also
- Old English at Wikipedia
- Old English in Middle-earth Part One and Part 2 at www.tolkiensociety.org
- James W. Bright's Glossary of the Anglo-Saxon Language at Jrrvf.com
- Sound samples of Old English verses related to Tolkien in Glǽmscrafu
- The Lord of the Rings and its Old English Origins: The Bones in the Soup by Alaric Hall
- Tolkien's Old English (forum thread)