Old English heh (Anglian), heah (West Saxon) "of great height, tall, conspicuously elevated; lofty, exalted, high-class," from Proto-Germanic *hauha- (source also of Old Saxon hoh, Old Norse har, Danish høi, Swedish hög, Old Frisian hach, Dutch hoog, Old High German hoh, German hoch, Gothic hauhs "high;" also German Hügel "hill," Old Norse haugr "mound"). The group is of uncertain origin; perhaps related to Lithuanian kaukara "hill," from PIE *kouko-. Spelling with -gh represents a final guttural sound in the original word, lost since 14c.
Of sound pitch, late 14c. Of roads, "most frequented or important," c. 1200 (high road in the figurative sense is from 1793). Meaning "euphoric or exhilarated from alcohol" is first attested 1620s, of drugs, 1932. Sense of "proud, haughty, arrogant, supercilious" (c. 1200) is reflected in high-handed and high horse. Of an evil or a punishment, "grave, serious, severe" (as in high treason), c. 1200 (Old English had heahsynn "deadly sin, crime").
High school "school for advanced studies" attested from late 15c. in Scotland; by 1824 in U.S. High time "fully time, the fullness of time," is from late 14c. High noon (when the sun is at the meridian) is from early 14c.; the sense is "full, total, complete." High finance (1884) is that concerned with large sums. High tea (1831) is one at which hot meats are served. High-water mark is what is left by a flood or highest tide (1550s); figurative use by 1814.
High and mighty is c. 1200 (heh i mahhte) "exalted and powerful," formerly a compliment to princes, etc. High and dry of beached things (especially ships) is from 1783.