late 14c., "praise" (a sense now obsolete), from Old French aloance "sanction, granting, allocation," from aloer "allot, apportion, assign" (see allow). As with allow, the English word involves senses of two different French words.
Meaning "sanction, approval, tolerance" is from 1550s. Sense of "a sum allotted to meet expenses" is from c. 1400. In accounts, meaning "a sum placed to one's credit" is attested from 1520s. Mechanical meaning "permissible deviation from a standard" is from 1903. To make allowances is literally to add or deduct a sum from someone's account for some special circumstance; figurative use of the phrase is attested from 1670s.