c. 1200, "fact of being born;" mid-13c., "act of giving birth, a bringing forth by the mother, childbirth," sometimes in Middle English also "conception;" also "that which is born, offspring, child;" from a Scandinavian source such as Old Norse *byrðr (replacing cognate Old English gebyrd "birth, descent, race; offspring; nature; fate"), from Proto-Germanic *gaburthis (source also of Old Frisian berd, Old Saxon giburd, Dutch geboorte, Old High German giburt, German geburt, Gothic gabaurþs), from PIE *bhrto past participle of root *bher- (1) "to carry; to bear children" (compare bear (v.)).
Suffix -th is for "process" (as in bath, death). Meaning "condition into which a person is born, lineage, descent" is from c. 1200 (also in the Old English word). In reference to non-living things, "any coming into existence" is from 1610s. Birth control is from 1914; birth certificate is from 1842.
mid-14c., actif, active, "given to worldly activity" (opposed to contemplative or monastic), from Old French actif (12c.) and directly from Latin activus, from actus "a doing" (from PIE root *ag- "to drive, draw out or forth, move").
As "capable of acting" (opposed to passive), from late 14c. Meaning "energetic, lively" is from 1590s; that of "working, effective, in operation" (opposed to inactive) is from 1640s. The grammatical active voice is recorded from 1765; grammatical use of active, signifying performance and not endurance of an action, dates from mid-15c. (opposed to passive or reflexive).
c. 1400, "active or secular life," from Old French activité, from Medieval Latin activitatem (nominative activitas), a word in Scholastic philosophy, from Latin activus "active" (see active). The meaning "state of being active, briskness, liveliness" is recorded from 1520s; that of "capacity for acting on matter" is from 1540s. As "an educational exercise," by 1923.