A separate master policy will cover the building itself and shared areas such as hallways and elevators, paid for by the co-op association. That cost is divided among all co-op owners as part of your dues.
The insurance you need is for your personal property (like that brand new furniture) and liability (what if your dog bites someone at your party?). In addition to your personal possessions, appliances and fixtures, your NYC co-op insurance policy can cover any renovations and upgrades you make to your unit, which can be a huge investment. Your mortgage lender may also require you have your own insurance, because your ability to repair your unit after a loss protects its value.