late 13c., "coarse cloth;" mid-14c., "tablecloth, bedspread;" from Old French carpite "heavy decorated cloth, a carpet" (Modern French carpette), from Medieval Latin or Old Italian carpita "thick woolen cloth," probably from Latin carpere "to card, pluck" and so called because it was made from unraveled, shredded, "plucked" fabric; from PIE root *kerp- "to gather, pluck, harvest." From 15c. in reference to floor coverings, which since 18c. has been the main sense. The smaller sort is a rug.
Formerly the carpet (usually in a single piece, like the Persian carpet) was also used (as it still is in the East) for covering beds, couches, tables, etc., and in hangings. [Century Dictionary]
From 16c.-19c., by association with luxury, ladies' boudoirs, and drawing rooms, it was used as an adjective, often with a tinge of contempt, in reference to men (as in carpet-knight, 1570s, one who has seen no military service in the field; carpet-monger, 1590s, a lover of ease and pleasure, i.e. one more at home on a carpet).
On the carpet "summoned for reprimand" is 1900, U.S. colloquial (but compare carpet (v.) "call (someone) to be reprimanded," 1823, British servants' slang). This may have merged with older on the carpet "up for consideration" (1726) literally "on the tablecloth," with the word's older sense, hence "a subject for investigation." To sweep or push something under the carpet in the figurative sense is first recorded 1953.