Old English (sometimes called Anglo-Saxon, though Tolkien disliked the term) was the ancestor language of modern English. It was also the mother language of characters Ælfwine and Eriol.
Anglo-Saxon used Runes for their writing system.
Tolkien was a professor of Anglo-Saxon at at Pembroke College from 1925 to 1945. He favored especially the Mercian dialect and was strongly influenced by Old English literature such as Beowulf.
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 although they are commented and edited by Christopher Tolkien, the translation is not given.
In earlier notes concerning the Book of Lost Tales, he commented that Old English was the only language the Elves of Eressëa could talk to Men, and that was how they talked to Ælfwine.
Anglo-Saxon runes were used on the Thrór's Map in The Hobbit to display the Cirth as used by the Dwarves.
Tolkien devised a Tengwar mode for Old English, for texts written by Edwin Lowdham in The Strange Case of Arundel Lowdham (DTS50 and DTS51).
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.