Consider a
ski slope and a set of
frictionless skis. If the hill is curved such that you can start at any point, ski straight down, and always arrive at the bottom in time T seconds, it is shaped like a brachistochrone.
Or consider a rabbit and a dog, both held in check. The rabbit is released and runs in a straight line at speed v, perpendicular to the initial path of the dog. When the dog is released, if it runs faster than the rabbit and always runs towards the rabbit, then the dog's path follows the curve y=c*f(x), where c is constant and f(x) is a brachistochrone.
Or consider a boat, some distance away from a pier. You are on the pier, and have a rope that leads to the boat. If you walk along the pier holding the rope, the path the boat will make as it is pulled by the rope is a brachistochrone.
The brachistochrone was first described by Bernoulli, and deriving it analytically was quite a tricky problem until Isaac Newton finally got it after inventing calculus. We now know that it is equal to one half-cycle of a cycloid, which is the path taken by a piece of gum stuck to the outside of a tire.
Brachistochrone is commonly misspelled as 'brachistochrome'.