Such money might be used as an undisguised bribe or to secretly promote the campaign of a politician. It might be money used by a CEO to buy luxury items or expensive vacations. Sometimes an officer of a charity might divert donations to personal use.
Two notorious examples are the Richard Nixon 1962 slush fund, which led to the famous “Checkers Speech,” and the Southern Methodist University football scandal, which totally wiped out their 1987 schedule.
The origin of the term involves an interesting bit of naval history. Slush was the fat and grease skimmed off when cooking meat. It was saved and used in later cooking as a type of lard. Excess amounts were sold to civilians when the ship reached port, and the money was used to buy things for the crew.
The Oxford English Dictionary gives this 1839 example by William McNally [Evils & Abuses in Naval & Merchant Service, xvii. 162]: “The sailors in the navy are allowed salt beef . . . From this provision, when cooked . . . nearly all the fat boils off; this is carefully skimmed . . . and put into empty beef or pork barrels, and sold, and the money so received is called the slush fund.”