Chess has not always looked like it does today. Over time the pieces and the rules have changed drastically. The modern game did not even emerge until the 1400's at least a millennium after it had been invented.
Chess is believed to have been invented in India, sometime before 500 AD. It quickly spread to Persia where it was called Shatranj, a Persian version of a Sanscrit word. It was adopted by the Muslims who retained the Persian name when they invaded. By 700 AD a version of the game had also reached mainland china where it was called Xiangqi, another modification of an Indian word.
In Europe, the game started to take on its current form around 1200 AD, and by 1475 it looked very much like the modern game we play. These standard rules developed in southern Italy, or possibly in Spain, but they spread throughout Europe rather quickly.
The standardization of this game which had for centuries given rise to variations was due to a number of factors. First, these rules were developed during the renaissance, a time of unprecedented learning and sharing of information. In this climate, the philosophy of chess developed, with strategies and mathematical analysis of the game becoming relatively widespread. In order to be sure that someone in England was theorizing about the same game as someone in Germany, a standard set of rules had to be agreed upon. In this way the development of the standard chess set is due largely to the development of chess theory.
This effect was furthered by the fact that the printing press was invented in the 1430's. With printing now easy and affordable treatises on chess and strategy could be easily spread throughout Europe. This led to the development of tournaments and eventually rankings, which also helped to solidify chess.
When Europe began to colonize large parts of the rest of the world, they brought their version of chess with them. In this way they took a game which was invented in India, and which had been played in china and the Middle East for hundreds of years, and reintroduced a new version to these cultures.
Today there are many different types of chess sets, but in general the rules are the same throughout the world. There are some variations such as Chinese checkers, but in general these are designated under different names, and chess means a game played by a standard set of rules.