It's easy to think that the belt is mainly the function of keeping trousers from falling down, but it has served many purposes historically. Belts have been worn, by both sexes, off and on starting as far back as 5000 years ago, probably longer. It's fair to assume that function was at the heart of the belt's conception, but probably more from the need to bring stuff with you without too much trouble than the need for keeping your trousers up. Belts have been predominantly a military or craftsman affair before the 20th century, although as with all things worn, sooner or later aesthetics will always play a certain role.
The modern use came about early in the 20th century when trouser waists fell to sit lower on the body and trousers started having the loops needed to attach the belt properly. Before that suspenders (braces) were more common for saving ill-fitted trousers for gravity. In modern times there are examples of when the function of keeping trousers up seems immaterial, for instance when trends have trousers sitting half way down the buttocks, and the belt hardly fills any upholding purpose whatsoever. Safe to assume aesthetics is the primary purpose. So it's easy to routinely say that the belt is positioned at the waist, but the truth is not that simple. The belt's position follows the fashion trends. In the 2nd half of the 20th century, the waist was probably a more accurate description of the belt's position than later, but it has and will probably be placed both higher and lower than that.
As mentioned, the belt's primary purpose in more ancient history has been to hang various items, pouches, weapons, money etc. But to a great extent it has also had an aesthetic purpose, and that was to accentuate the waist, which probably came quite naturally since historic belts were worn on top of coats rather than in trousers, mainly military uniforms or craftsmen's clothing. The aesthetic aspect in the military went so far as to tighten the belts to physically help the soldiers to a better posture, to convey fitness and discipline (an abdominal contraction which, to be honest, must have been detrimental to their fighting abilities :) Despite what you might think, this nonsensical practice has actually followed humanity into modern days.
The belt consists of a strap that goes around the waistband, usually of leather, textile or a synthetic material. The strap has a number of holes to make the length adaptable to the wearer's body size. The strap goes around the waistband of the trousers, through several loops holding the belt in place, and finally across the front, into a buckle made of some hard material, usually metal. Finally the belt is fastened by the prong, the little bar attached to the buckle, which goes into the hole that gives the belt the wanted tightness. The point of the belt sometimes has a metal piece called the end tip.