oratory n.1

late 14c., oratour, "an eloquent or skilled speaker; one who pleads or argues for a cause," from Anglo-French oratour (Modern French orateur) and directly from Latin orator "speaker," from ōrare "to speak, speak before a court or assembly, pray to, plead."

This is sometimes said to be  from PIE root *or- "to pronounce a ritual formula" (source also of Sanskrit aryanti "they praise," Homeric Greek are, Attic ara "prayer," Hittite ariya- "to ask the oracle," aruwai- "to revere, worship").  But according to de Vaan, the Latin word is rather from Proto-Italic *ōs- "mouth," from PIE *os- "mouth" (see oral). He writes:

The chronology of the attestations shows that 'to plead, speak openly' is the original meaning of orare .... The alternative etymology ... seems very unlikely to me: a connection with Skt. a-aryanti 'they acknowledge' and Ru. orat' 'to shout', since nothing suggests a meaning 'to shout' for the Latin verb, nor does it seem onomatopoeic.

The general meaning "public speaker," is attested from early 15c. Fem. forms were oratrice (early 15c., from Anglo-French); oratrix (mid-15c., from Latin); oratress (1580s).

"long musical composition, usually with a text based on Scripture," 1727 (in English from 1640s in native form oratory), from Italian oratorio (late 16c.), from Church Latin oratorium "a place of prayer, an oratory or chapel," noun use of an adjective, as in oratorium templum, from neuter of Latin oratorius "of or for praying," from ōrare "to pray, plead, speak" (see orator). The purely musical sense stems from the Oratory musical services of prayers and hymns instituted in Rome in the 1550s by St. Philip Neri and performed in the church of Santa Maria in Vallicella.

"pertaining to an orator or oratory," 1580s, from orator or oratory + -ical, or else from Latin oratorius "pertaining to an orator or to speaking or pleading" (see oratory (n.1)). Related: Oratorically. Earlier form was oratorial (1540s).