Fine art is a major investment, some pieces can be worth millions of dollars. Unfortunately some times art gets stolen or damaged. In this case you will be very glad you have insurance to cover the cost. It is true that many pieces of art can't be replaced and often insurance money is small compensation for the loss of a valuable piece of art, but it is better than receiving nothing. A lot of people choose not to get art insurance because of the cost, this is a mistake, the cost of the insurance is nothing compared to the cost of losing a piece of art.
Art insurance is pretty much what you would think it would be, it is insurance to protect your art. There are actually two different ways that you can get art insurance depending on your situation. If you have art in your home you can add the art to your homeowners insurance policy. In general artwork, particularly valuable artwork won't be covered by your homeowners policy. It is usually possible to add coverage for your art, for a fee of course, in order to protect yourself if the art is damaged or stolen. If you are going to be displaying your art outside of your home you will need to get an insurance policy specifically to cover your art. Given the cost of many pieces of art the insurance can be very expensive but it is almost always a good idea.
Art insurance for museums and galleries in particular can be a problem especially if they have very valuable pieces. Given the cost of the insurance and the fact that most museums and galleries don't have a ton of money being able to afford the premiums is usually a problem. In this case the best option is to buy as much insurance as you can and make sure that the most valuable pieces are covered. If some of the lower valued pieces of art are left without insurance coverage that is a risk you may just have to take but it is a risk you can't afford with a valuable piece.
It is important that when you buy art insurance you make sure that you understand exactly what is covered, and for what. Loads of art galleries (even the good ones) offer insurance policies these days, yet have very little knowledge of what they include (and what they don't include). If there is something in your art insurance policy that you don't understand you need to get it clarified. It is also important that you make sure that you photograph all of your art and have appraisals done on a regular basis. You need to be able to show the insurance company an accurate appraisal of the value of the art if you want to be compensated properly. Insurance companies try to limit the amount that they payout, they will almost certainly undervalue your art unless you can show them an appraisal that says otherwise.