digraph representing a sound found chiefly in words of Old English, Old Norse or Greek origin, but unpronounceable by Normans and many other Europeans. In reconstructed PIE origins, the Greek -th- and the Germanic -th- descend from different sound roots.
In Greek, -th- at first represented a true aspirate (T + H, as in English outhouse, shithead, etc.). But by 2c. B.C.E. the Greek letter theta was in universal use and had the modern "-th-" sound.
Latin had neither the letter nor the sound, and the Romans represented Greek theta by -TH-, which they generally pronounced, at least in Late Latin, as simple "-t-" (passed down to Romanic languages, as in Spanish termal "thermal," teoria "theory," teatro "theater").
In Germanic languages it represents a sound common at the start of words or after stressed vowels. To indicate it in alphabetic writing, Old English and Old Norse used the characters ð "eth" (a modified form of -d-) and þ "thorn," which had been a rune. Old English, unlike Old Norse, seems never to have standardized which of the two letters represented which of the two forms of the sound ("hard" and "soft").
The digraph -th- sometimes appears in early Old English writing, on the Latin model, and it returned in Middle English with the French scribes, driving out eth by c. 1250, but thorn persisted, especially in demonstratives (þat, þe, þis, etc.), even as other words were being spelled with -th-.
The advent of printing dealt its death-blow, however, as the first types were imported from continental founders, who had no thorn. For a time y was used in its place (especially in Scotland), because it had a similar shape, hence ye for the in pseudo-historical typographical affectation Ye Olde _____ (it never was pronounced "ye," only printed that way; see ye (article)).
After the Renaissance, English writers saw that some words inherited from French or Latin with a t- had been th- in the original Greek. The -th- was restored in amethyst, asthma, pythoness, orthography, theme, throne, etc.); it failed in acolyte. Over-correction in English created unetymological forms such as Thames and author. Caxton (late 15c.) has thau for tau, and compare Chaucer's Sir Thopas (topaz). The earliest form of Torah in English was Thora (1570s). Yet some words borrowed from Romanic languages preserve, on the Roman model, the Greek -th- spelling but the simple Latin "t" pronunciation (Thomas, thyme).