O.E. oþer "the second, one of the two, other,"
from P.Gmc. *antharaz (cf. O.S. athar, O.N. annarr, Ger. ander, Goth. anþar "other"),
from PIE *an-tero-, variant of *al-tero- "the other of two" (cf. Lith. antras, Skt. antarah "other, foreign," L. alter),
from base *al- "beyond" + adj. comp. suffix *-tero-.
Sense of "second" was detached from this word in English (which uses second, from Latin) and German (zweiter, from zwei "two") to avoid ambiguity.
In Scandinavian, however, the second floor is still the "other" floor (cf. Swed. andra, Dan. anden).
Phrase other world "world of idealism or fantasy, afterlife, spirit-land" is c.1200;
hence otherworldliness (c.1834).
The other woman "a woman with whom a man begins a love affair while he is already committed" is from 1855.
The other day originally (mid-12c.) was "the next day;"
later (c.1300) "yesterday;" and now, loosely, "a day or two ago" (early 15c.).
Phrase other half in reference to either the poor or the rich, is recorded from c.1600.
La moitié du monde ne sçayt comment l'aultre vit. [Rabelais, "Pantagruel," 1532]