c. 1200, "the face, countenance," especially as expressing emotion, from Anglo-French chere "the face," Old French chiere "face, countenance, look, expression," from Late Latin cara "face" (source also of Spanish cara), possibly from Greek kara "head," from PIE root *ker- (1) "horn; head." From mid-13c. as "frame of mind, state of feeling, spirit; mood, humor."
By late 14c. the meaning had extended metaphorically to "state or temper of mind as indicated by expression." This could be in a good or bad sense ("The feend ... beguiled her with treacherye, and brought her into a dreerye cheere," "Merline," c. 1500), but a positive sense, "state of gladness or joy" (probably short for good cheer), has predominated since c. 1400.
Meaning "that which makes cheerful or promotes good spirits" is from late 14c. Meaning "shout of encouragement" is first recorded 1720, perhaps nautical slang (compare earlier verbal sense, "to encourage by words or deeds," early 15c.). The antique English greeting what cheer? (mid-15c.) was picked up by Algonquian Indians of southern New England from the Puritans and spread in Indian languages as far as Canada.
late 14c., "to humor, console, dispel despondency;" c. 1400 as "entertain with food or drink," from cheer (n.). Related: Cheered; cheering. Sense of "to encourage by words or deeds" is early 15c. Which had focused to "salute with shouts of applause" by late 18c. Cheer up (intransitive) first attested 1670s.