For the extensive damage to your own vehicle you will need to make a claim with your insurance company. This type of claim is normally a collision claim since your vehicle collided with an object (the fence) and thus you will need to pay the deductible which you chose when you started your collision coverage.

As for the damage to the fence, the owner will likely make a claim through your property damage liability coverage. PD liability covers you if your car damages someone else's property such as their car or fence or other property.

Seven Steps to Filing Your Insurance Claim is an article in our learning center that also might be helpful as you file your claim for your vehicle's damages.