It's a competitive, frustrating routine which seems endless. It's usually to gain wealth and/or power.

A fierceful competetive life style to make better social status.

It's basically modern life in a big city. When people say that they are going on holidays to escape the 'rat race' they are going away to escape the stress of modern life. When they move to the quiet countryside or move to a remote, relaxing place to live to get out of the 'rat race' they are mere trying to avoid the fast-paced, stressful lifestyle in the inner cities. (Just did it myself!)

I think the phrase emerged in the 1950s. It means (or originally meant) "the competitive struggle for status within a large company." It probably was a reference to psychological research. Psychologists put laboratory rats in mazes, and time how long it takes for them to find their goal. This was a standard research tool from about 1900 to 1940 or so. They were literally racing rats. Rats racing through mazes is a popular metaphor. It is used for human activity that feels frustrating, artificially constrained, and unfulfilling. (The rat doesn't really achieve anything. They get a bit of cheese, but they just get put back and have to run the same maze over and over and over). This metaphor is behind a recent best-selling motivational book about being adaptable, "Who Moved My Cheese," by Spencer Johnson. Another frequently-used metaphor is the "hamster wheel." Hamsters and other pet rodents are often kept in cages with circular wheels. They like to climb into the wheel and run. The wheel turns and turns. They apparently enjoy it but they don't get anywhere. People will say "I feel like I'm in a hamster wheel" if they are, metaphorically, running hard but not getting anywhere.