Young men from richer families who didn't have to work in most Greek cities spent a lot of their time training for these competitions, and the best of them were chosen to compete against the best young men from other cities. Then they would all meet, at the Olympic Games or the Isthmian Games or elsewhere, and compete for prizes and for the favor of the gods. Of course these games also served as good training for the army, because all these men would be soldiers as well. The events were the same kind as in the Olympics today: running, jumping, throwing a javelin, and throwing a discus. Only men could compete.
While yes indeed, Greece is proud to be the host for the 2004 games. Greece is also proud of its hertiage of being the creators of the Modern Olympic games. Check out this great page on the first Modern Olympic Games.
The revival of the Modern Olympics was the brain child of the Frenchman, Baron Pierre de Coubertin. He believed that international competitions between amateur athletes would help promote friendly relationships between people from different countries. Despite strong opposition in 1894 de Coubertin assembled 79 delegates from 12 countries to attend the international congress for the re-establishment of the Olympic Games. It was decided to hold the first modern Olympics in Athens in two years time.