Meaning "entertainment or public reception as a greeting" is recorded from 1530. The adjective is from Old English wilcuma. You're welcome as a formulaic response to thank you is attested from 1907. Welcome mat is from 1908; welcome wagon is attested from 1940.
*gwā-, also *gwem-, Proto-Indo-European root meaning "to go, come."
It forms all or part of: acrobat; adiabatic; advent; adventitious; adventure; amphisbaena; anabasis; avenue; base (n.) "bottom of anything;" basis; become; circumvent; come; contravene; convene; convenient; convent; conventicle; convention; coven; covenant; diabetes; ecbatic; event; eventual; hyperbaton; hypnobate; intervene; intervenient; intervention; invent; invention; inventory; juggernaut; katabatic; misadventure; parvenu; prevenient; prevent; provenance; provenience; revenant; revenue; souvenir; subvention; supervene; venire; venue; welcome.
It is the hypothetical source of/evidence for its existence is provided by: Sanskrit gamati "he goes," Avestan jamaiti "goes," Tocharian kakmu "come," Lithuanian gemu, gimti "to be born," Greek bainein "to go, walk, step," Latin venire "to come," Old English cuman "come, approach," German kommen, Gothic qiman.
"worthy to be desired, fit to excite a wish to possess," late 14c., from Old French desirable (12c.), from desirrer (see desire (v.)). In Middle English sometimes it meant "desired, hoped for, welcome." Related: Desirably.
1530s, "a cleaning out, removal, clearance," from rid + -ance. The meaning "a deliverance from something superfluous or unwanted" is from 1590s. Good riddance, "a welcome relief from unpleasant company or an embarrassing connection" attested from 1650s. Shakespeare has gentle riddance (1590s); Middleton has fair riddance (1610s).
From 1560s in a physical sense of "thrust in" (transitive or intransitive); meaning "enter unbidden and without welcome" is from 1570s; that of "thrust or bring in without necessity or right" is from 1580s. Related: Intruded; intruding.
c. 1300, receiven, "take into one's possession, accept possession of," also in reference to the sacrament, from Old North French receivre (Old French recoivre) "seize, take hold of, pick up; welcome, accept," from Latin recipere "regain, take back, bring back, carry back, recover; take to oneself, take in, admit," from re- "back," though the exact sense here is obscure (see re-) + -cipere, combining form of capere "to take" (from PIE root *kap- "to grasp").
From c. 1300 as "welcome (in a specified manner)." From early 14c. as "catch in the manner of a receptacle." From mid-14c. as "obtain as one's reward." From late 14c. as "accept as authoritative or true;" also late 14c. as "have a blow or wound inflicted." Radio and (later) television sense is attested from 1908. Related: Received; receiving. Receiving line is by 1933.
Other obsolete English verbs from the same Latin word in different forms included recept "to receive, take in" (early 15c., recepten, from Old French recepter, variant of receter and Latin receptus). Also compare receipt, which also had a verb form in Middle English, receiten.