1. Now that Mr. Anderson is rich, he rides around town in a fancy car like a fat cat and has everyone else do the work.
2. Richard is such a fat cat businessman. He owns several supermarkets.

The expression is primarily used to describe a man (usually not a woman) who, although rich, is not seen as industrious. He has perhaps become wealthy through hard work but is now relaxing, or he has never had to work for his wealth.