Idiomatically, "a fat cat" is someone who is very wealthy and, as a result, is able to eat more food than necessary and otherwise enjoy a life of luxury. Thus, they resemble fat, lazy cats that eat, sleep and do nothing useful.
Once, wealthy people who were plump (not obese necessarily, but merely larger than average) had their size seen as proof of their wealth, showing that they could survive famine and that they stood above ordinary people.
Today, obesity is more of a problem with poorer people who cannot afford health food, gym memberships, and all the various trappings of a "healthy lifestyle." Fatter foods have become cheaper and available to the masses, while healthier foods have been made more expensive.
In practice, any wealthy person can be described as "a fat cat," but in modern times, this has a clear, negative connotation.
Example: "The fat cat who runs the local bank is a greedy jerk who enjoys taking advantage of people. Even though he has so much money, he tries to cheat other people in business, even over the most trivial things! He won't even tip waiters at restaurants properly." This describes a person viewed (by the speaker) as being unjustly rich, and unjustly aggressive in seeking to save trivial amounts of money (to a rich person).