Sometimes chasing the origin and meanings of an apparently simple English word gives us a glimpse of a bigger history, as well as a reminder on spelling.
Curb has probably come to us, in most of its senses at least, from the French courbe, a word which was also used directly in English, which in turn comes from the Latin curvus, meaning bent or crooked, something which is far removed from almost all of its modern meanings.
As a noun, a curb is an additional piece fastened to the upper ends of the branches of a horse’s bit, passing under the lower jaw, so as to check an unruly horse, or one which fights the bit. The OED documents that use as far back as 1477, and shows it extending to mean a check or constraint more generally by the early seventeenth century.
An apparently separate use is recorded almost as early, to describe a hard swelling on the hock or other part of the leg of a horse.
Next, by 1800, the OED finds it used for a mould or template for curved constructions, particularly in architecture and building work.
It also had connotations with curved outlines which have extended quite separately from the meanings above, to cover an enclosed framework or border, including the coaming round the top of a well (as early as 1511), framing round the top of a brewer’s copper, an opening in a floor or roof, and other circular plates or cylinders of wood or metal.
With the advent of road construction in the early nineteenth century, a margin of stone became used to segregate pedestrians from vehicles, and that too was termed a curb. Whilst it has retained that original spelling in the US, here in the UK it changed quite quickly into kerb.
In the latter half of the nineteenth century, it was also applied to a now obsolete practice in telegraphy, a technique for preventing signal delay during cable transmission.
Some of these nouns have become verbalised too. Although you are not likely to hear curb used to mean to bend or curve an object, it remains in common use for to check or keep in check. It also sometimes means to provide with a curb or kerb, spelled according to nationality.
In its British English kerb form, it is popular in some compounds, such as kerb-merchant for a street salesman, and kerb-crawling for the act of driving round slowly in a car to pick up a prostitute.
So curb on a horse, your desires, or the American sidewalk, but kerb on the British pavement: curves that first checked and constrained before going (mainly) straight.