straight (adj.1) Look up straight at Dictionary.com
late 14c., "direct, undeviating; not crooked, not bent or curved," of a person, "direct, honest;" properly "stretched," adjectival use of Old English streht (earlier streaht), past participle of streccan "to stretch" (see stretch (v.)). Related: Straightly; straightness.

Meaning "true, direct, honest" is from 1520s. Of communication, "clear, unambiguous," from 1862. Sense of "undiluted, uncompromising" (as in straight whiskey, 1874) is American English, first recorded 1856. As an adverb from c.1300, "in a straight line, without swerving or deviating." Theatrical sense of "serious" (as opposed to popular or comic) is attested from 1895; vaudeville slang straight man first attested 1923.

Go straight in the underworld slang sense is from 1919; straighten up "become respectable" is from 1907. To play it straight is from 1906 in theater, 1907 in sports ("play fair"), with figurative extension; later perhaps also from jazz. Straight arrow "decent, conventional person" is 1969, from archetypal Native American brave name. Straight shooter is from 1928. Straight As "top grades" is from 1920.
straight (adj.2) Look up straight at Dictionary.com
"conventional," especially "heterosexual," 1941, a secondary sense evolved from straight (adj.1), probably suggested by straight and narrow path "course of conventional morality and law-abiding behavior," which is based on a misreading of Matt. vii:14 (where the gate is actually strait), and the other influence seems to be from strait-laced.
straight (n.) Look up straight at Dictionary.com
1640s, "a level position," from straight (adj.1). From 1864 as "straight part of a race track." Poker sense attested from 1841. Meaning "conventional person" is first recorded 1967, from straight (adj.2).