In general most auto insurance policies will provide liability coverage for trailers if you own it. Many auto insurance policies do extend this type of coverage to trailers if they are being pulled by your insured vehicle at the time of an accident. We must point out we cannot say for certain if your policy would cover a trailer since state laws and insurance company guidelines and policy terms vary greatly.
Whether other parts of an auto insurance policy, besides your liability coverages, would cover a trailer while it is being towed would depend upon the terms of your policy and guidelines of your insurance carrier. In many cases the trailer would need its own comprehensive or collision coverage. Some insurers would require the trailer to be listed on the declarations page of your policy to be covered.
Trailer coverage will vary by state and by insurance company. Some will extend liability coverage while its attached to your motor vehicle while others will say to be covered you must have a separate policy for the trailer or an endorsement to your personal auto insurance to cover the trailer.
It could also depend upon the type of trailer you are towing. If it is a utility trailer with a car on it or a pleasure craft such as an ATV or watercraft than it likely will be extended coverage though you would need to check about coverage for the contents of the trailer. If it is a travel trailer it may need its own policy on it to be covered.
If your trailer is covered by your personal auto insurance policy you should also ask your agent if the contents or items being towed on the trailer are covered. It may be that the contents would not be covered under your car insurance without an endorsement even if your trailer is covered.
Auto insurance policies will not typically cover a rented trailer though. If you get a trailer from U-haul for instance than you normally would need to also get their coverage on the trailer since rented equipment such as this usually is not covered by your personal auto insurance or any of the insurance benefits that credit cards sometimes extend you for say driving a rental car.
If you want to know if your current car insurance policy or a policy that you are going to purchase for your vehicle will extend to a trailer that you pull with your insured vehicle read through the terms of your policy and ask the insurance provider for any clarification you need. Ask if there are any size and/or valuation limitations if the trailer is not listed and separately insured.