There are many types of trailers that you can pull behind your car, SUV or pickup truck. Most car insurance will extend liability coverage to the trailer being pulled by your vehicle.
However, if the pulling vehicle is in commercial use, such as a contractor or farmer, and you have commercial vehicle insurance on the towing vehicle, liability MAY NOT extend to the trailer being towed. This is something that you definitely want to verify with your insurance company before hitching up. Coverage may be able to be provided by endorsement to your policy.
Insurance for items ON the trailer is not provided by your car insurance however. For example, if you are towing a trailer full of household items when moving and the trailer becomes unhitched and rolls, coverage for the household items would have to come from your homeowners or renters insurance. Your auto insurance will only pay for damage to objects destroyed by the trailer when it rolled. It does not provide coverage for the trailer itself however. You would need to insure the trailer specifically in order to have coverage.
For the average driver who has personal car insurance and pulls a utility trailer or camper, liability coverage will extend from the pulling vehicle. Your homeowners policy may provide a small amount of physical damage coverage for the trailer. Again, check with your insurance company or agent if you are not sure about your coverages.