"either, else, otherwise, as an alternative or substitute," a phonetically reduced form (analogous to e'er from ever) of Middle English other, a conjunction joining two or more alternatives, from Old English oþþe "either, or."
Oþþe was extended in early Middle English (and Old High German) with an -r ending, perhaps by analogy with other "choice between alternative" words (either, whether). The contraction changed the second term in an alternative expression such as either ... or, which is from a common construction in Old English, where both words originally were oþþe (compare nor).
Midlands dialect had the condensed form or by mid-13c., and it was common in texts by c. 1300, especially in northern parts of England. The fuller form other for "or" continued in the South into 16c.
Tell us by what auctorite thou doest these thynges? Other who is he that gave the thys auctorite? [Luke xx, Tyndale, 1526]
In Middle English it also was confused or merged with outher, ouþer (pron.) "either one, either of two or more," from Old English æghwæðer, the source of either. This, too, probably was by association in such phrases of alternative condition as either ... or.
Germanic cognates of Old English oþþe include Old Frisian ieftha, Middle Dutch ofte, Old Norse eða, Old High German odar, German oder, Gothic aiþþau "or."
Or else "otherwise" is by c. 1300.