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.

1868, from normal (in reference to molecular structure) + epinephrine.

1875, from sub- + normal. The noun is from 1710 in geometry; 1916, of persons.

"character or state of being in accordance with rule or standard," 1833, from normal + -ity.

"reduce to a standard; cause to conform to a standard," 1848, from normal + -ize. Related: Normalized; normalizing.

1590s, "regularly, according to general custom" (a sense now archaic or obsolete), from normal + -ly (2). Meaning "under ordinary conditions" is by 1838.

"establishing or setting up a norm or standard which ought to be followed," 1880, perhaps from French normatif, from Latin norma "rule" (see normal).

1905, in reference to observed events or things presumed to operate by natural laws but not conforming to those known or normal, from para- (1) + normal. Related: Paranormally.

1857, "mathematical condition of being at right angles, state or fact of being normal in geometry," from normal + -cy. The word has been associated since 1920 with U.S. president Warren G. Harding (who campaigned that year under the slogan "Return to Normalcy," meaning pre-World War I conditions). Previously normalcy was used mostly in the mathematical sense and the word preferred by purists for "a normal situation" is normality. Harding's use of it was derided during his administration as an example of his much-belittled incompetence with the language (Democratic politician William G. McAdoo Jr. called Harding’s speeches "an army of pompous phrases moving across the landscape in search of an idea"). 

America’s present need is not heroics, but healing; not nostrums, but normalcy; not revolution, but restoration; not agitation, but adjustment; not surgery, but serenity; not the dramatic, but the dispassionate; not experiment, but equipoise; not submergence in internationality, but sustainment in triumphant nationality. [Harding, "Readjustment" speech, May 24, 1920]

"act or process of making normal," 1842, from normalize + -ation. International political sense "achieving of stable relations between two countries" is recorded by 1938.