c. 1500, "typical, common;" 1640s, in geometry, "standing at a right angle, perpendicular," from Late Latin normalis "in conformity with rule, normal," in classical Latin "made according to a carpenter's square," from norma "rule, pattern," literally "carpenter's square," a word of unknown origin (see norm). Meaning "conforming to common standards or established order or usage, regular, usual" is attested from 1828 but probably is older than the record [Barnhart].
Meaning "heterosexual" is by 1914. As a noun meaning "usual state or condition," from 1890 (in geometry as "a perpendicular" from 1727). Sense of "a normal person or thing" is attested by 1894. Normal school "training college for teachers" (1835) is a translation of French école normale (1794), a creation of the French Republic; the notion is of "serving to set a standard." The U.S. city of Normal, Illinois, was named 1857 for the normal school established there.