Your question is too broad for a simple yes or no answer. Depending on the circumstances, your appliances may be covered under your homeowners insurance, but other problems might be better suited to a home warranty. The important thing is to know where your home insurance leaves off and a home warranty becomes more useful. For maximum financial protection, it might be a good idea to carry both.
Homeowners insurance is to protect you against loss. Home insurance will cover the home, other buildings on the property, and your personal property (including appliances) from theft, vandalism, and many types of disaster or destruction. Make sure that your insurance company has an up to date copy of your home inventory, and that all of your home appliances, even your lawnmower, are included. In the event of a total loss, you will be able to settle the claim faster if you have proof of the items you have lost.
A home warranty is like an expanded warranty for appliances in the home. It will not replace items destroyed or damaged in a catastrophe, or items that are stolen from the property. If your refrigerator is on the home warranty and the compressor goes out, your home warranty will pay for it to be repaired or replaced at little or no charge to you. As you can see, a home warranty isn't much like an insurance policy at all; it is really more of a maintenance contract for your assorted home appliances.