The C-round for Civilians is spent investing in setting up their own businesses. At the beginning of the game it's unlikely you'll be able to afford more than a stall in the street, but if you make enough money, or side with the right mafia Family then you might soon end up owning a chain of hotels and restaurants across Central Manhattan.
Other, jealous Civilians may want to ruin your business by breaking into your properties and stealing money. They could put in requests for such action in the B-round, although they can also do it informally during the A-round. Muggings are common, and the only way to avoid this (or at least get revenge afterwards) is to have allegiance to a mafia Family who will "sort out" the offenders on your behalf.
So you can see there are many advantages to siding with a Family. However, there are major drawbacks. You might get dragged into gang warfare, you might fall into disfavour, you might be extorted when they're short of cash. It's a risk you have to take...