It is never made completely clear EXACTLY what business they engage in, but it seems to involve finances (perhaps lending). Dickens refers to his place of business as a "counting house."
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Scrooge worked for Fezziwig when he was young, but his business partner was Jacob Marley.
Scrooge and Marley were merchants and ran warehouses. In additon they loaned money and retented property
Following Scrooges stated "you were always a good business man Jacob". Marley cries out almost in pain`Business! Mankind was my business. The common welfare was my business; charity, mercy, forbearance, and benevolence, were, all, my business. The dealings of my trade were but a drop of water in the comprehensive ocean of my business!''
There doesn't seem to be a specific reference to the type of business Scrooge owns. Since his only employee is an accountant, one might speculate that Scrooge is a money-lender.
Loans and debts.