Great question! It would depend on the coverage on your policy, but the liability portion of your policy normally extends to cover injury or damage that was caused to someone else by your trailer. If your auto policy also includes comprehensive and collision (full) coverage then your trailer may also be covered for damage. The amount is normally up to about $500 in value, but not all policies are exactly the same so you would need to check your policy or call your insurance company just to be sure.
It's also important to specify that the size and type of the trailer does matter. Many companies offer coverage specifically for travel and livestock trailers.