Proto-Indo-European root meaning "stranger, guest, host," properly "someone with whom one has reciprocal duties of hospitality," representing "a mutual exchange relationship highly important to ancient Indo-European society" [Watkins]. But as strangers are potential enemies as well as guests, the word has a forked path.
The word ghos-ti- was thus the central expression of the guest-host relationship, a mutual exchange relationship highly important to ancient Indo-European society. A guest-friendship was a bond of trust between two people that was accompanied by ritualized gift-giving and created an obligation of mutual hospitality and friendship that, once established, could continue in perpetuity and be renewed years later by the same parties or their descendants. [Calvert Watkins, "American Heritage Dictionary of Indo-European Roots"]
It forms all or part of: Euxine; guest; hospice; hospitable; hospital; hospitality; hospodar; host (n.1) "person who receives guests;" host (n.2) "multitude;" hostage; hostel; hostile; hostility; hostler; hotel; Xenia; xeno-; xenon.
It is the hypothetical source of/evidence for its existence is provided by: Greek xenos "guest, host, stranger;" Latin hostis, in earlier use "a stranger," in classical use "an enemy," hospes "host;" Old Church Slavonic gosti "guest, friend," gospodi "lord, master;" Old English gæst, "chance comer, a stranger."
Sense of "home for the aged and terminally ill " is from 1879; hospice movement first attested 1978.
"person who receives guests," especially for pay, late 13c., from Old French oste, hoste "guest, host, hostess, landlord" (12c., Modern French hôte), from Latin hospitem (nominative hospes) "guest, stranger, sojourner, visitor (hence also 'foreigner')," also "host; one bound by ties of hospitality."
This appears to be from PIE *ghos-pot-, a compound meaning "guest-master" (compare Old Church Slavonic gospodi "lord, master," literally "lord of strangers"), from the roots *ghos-ti- "stranger, guest, host" and *poti- "powerful; lord." The etymological notion is of someone "with whom one has reciprocal duties of hospitality" [Watkins]. The biological sense of "animal or plant having a parasite" is from 1857.
Spelling evolution influenced by Old Norse cognate gestr (the usual sound changes from the Old English word would have yielded Modern English *yest). Meaning "person entertained for pay" (at an inn, etc.) is from late 13c. Old English also had cuma "stranger, guest," literally "a comer." Phrase be my guest in the sense of "go right ahead" first recorded 1955.