The Traveling Salesman Problem is to find the shortest circuitous path connecting N cities (meaning that a traveling salesman following that path would visit each city only once). Although it can in principle be solved by brute force (by calculating the length of every possible circuit), this is not practical because the number of circuits grows so fast that even for N = 25 cities, it would take longer than the age of the universe (~10 billion years) to check every path, at a rate of one million paths per second!
However, the method of simulated annealing quickly gives a reasonable answer, where "reasonable" means close enough to the true minimum path for practical purposes. Simulated annealing starts with the cities connected in a random order, and then considers making random changes in that order. If changing the order of cities leads to a shorter path, we accept that change. If the modification yields a longer path, we give ourselves a certain probability of accepting the modification -- less likely the larger the proposed increase in path length. We then gradually reduce this probability over time, in order to rule out shorter and shorter path increases -- thereby converging toward a path length close to the absolute minimum.
The term "simulated annealing" comes from the analogy with annealing of metal, in which the metal is heated to a high temperature to give its atoms a lot of thermal motion, then is slowly cooled to give them a chance to align themselves into their lowest-energy (generally crystalline) configuration. Annealing makes the metal stronger than does rapid cooling (e.g. by plunging it into cold water), which would freeze the atoms wherever they happened to be, leaving microscopic cracks that make the metal brittle.
To make a new map (with the cities connected at random), enter the number of cities you desire, and then click Make New Map. To start the shortest-path calculation, click Start Annealing.