Rat race usually means what Andrew has answered. A way of life in which people are caught up in a fiercely competitive struggle for wealth or power. But it also means an exhausting routine that leaves no time for relaxation. A difficult, tiring, often competitive activity or routine.

The rat race means a way of life in modern society, in which people compete with each other for power and money.

It refers to the feeling that you are constantly struggling to improve and succeed, but then, when you accomplish your goals you simply have to do more. There is always something more that you need to do when you accomplish your objective. For example, someone wants a new car, then they get the new car and they feel they need a new house, then they get the new house, then they feel they need a new job that pays more, then when they get the new job they need to pay for college for their kids, then they need a new car again. It is called the "rat race" because it feels like (and looks like) the person is a rat who can never catch their prize.

A rat race is an endless, self-defeating, or pointless pursuit. It conjures up the image of the futile efforts of a lab rat trying to escape while running around a maze or in a wheel. http://en.wikipedia.org/wiki/Rat_race