Condo or Coop Insurance covers you for your Contents and Liability, much like a Homeowners Policy. However, when it comes time for a claim, determining who is a fault or negligent and who is responsible becomes difficult because you will have multiple insurance companies along with a set of By-Laws and probably even a management company.
Much like a Homeowners Policy, anything the falls down when you "flip" your house upside-down will be considered contents. This will be furniture, clothing, electronics, food, ext. You will usually choose this when purchasing a policy.
Also, much like a Homeowners Policy you have a liability section that covers you for lawsuits, property damage, and bodily injury. This can be quite a bit more complicated in a Condominium or Coop.
Betterments & Improvements
Anything you add to the physical structure. A new kitchen, bathroom tile, new toilet, sink, door, trim, light fixture, ceiling fan, paint. This is usually going to be covered under your building coverage.
A Cooperative is different legal entity where you are purchasing shares in a building and occupy an apartment per the proprietary lease. When dealing with claims you will need this.