late 14c., paiement, "action of paying, repayment of a debt; amount due as a payment," from Old French paiement (13c.), from paiier (see pay (v.)). Meaning "thing or sum of money given in discharge of a debt" is from mid-15c.
late 14c., pensioun, "payment for services," especially "a regular reward or annual payment out of a will or benefice" (early 14c., in Anglo-Latin), from Old French pension "payment, rent" (13c.) and directly from Latin pensionem (nominative pensio) "a payment, installment, rent," from past-participle stem of pendere "to hang, cause to hang; weigh; pay" (from PIE root *(s)pen- "to draw, stretch, spin"). For the financial sense of the Latin verb, see pound (n.1).
Meaning "regular payment to a person in consideration of past service" is from 1520s, hence "periodic payment made to a person retired from service on account of age or disability" (originally especially government pay to soldiers and sailors). Meaning "boarding house, boarding school" is attested from 1640s, from a sense in French based on the meaning "money paid for board," and in English it is usually in reference to places in France or elsewhere on the Continent.
"a partial payment on account of debt due," 1776, earlier "the arrangement of payment" (1732), an alteration of Anglo-French estaler "to fix payments," from Old French estal "fixed position, place; stall of a stable, market, or choir," from a Germanic source akin to Old High German stal "standing place" (see stall (n.1)). General sense of "a part of a whole, furnished or produced in advance of the rest" is from 1823. Installment plan is from 1894.
Modern meaning "exaction of money payment for an offense or dereliction" is via sense of "sum of money paid for exemption from punishment or to compensate for injury" (mid-14c., from the same sense in Anglo-French, late 13c.) and from phrases such as to make fine "make one's peace, settle a matter" (c. 1300). Meaning "sum of money imposed as penalty for some offense" is first recorded 1520s.