There are two main types of insurance you'll need to buy before close - title insurance and homeowners' insurance. Both of these will be required by your mortgage broker, and are generally a good idea to protect a very valuable asset.
Title insurance protects you in the case of defects or problems with the title of your new property. Title insurance is generally organized by the escrow agent, but paid for by the buyer. The buyer will also pay for their lender's title insurance.
Homeowners' insurance protects you from loss or damage to your new property or the possessions within the property. A standard homeowners' insurance policy will cover losses from events like fire or burglary, and may cover if someone is injured in your home. The buyer is responsible for purchasing homeowners' insurance directly, and you can get quotes from major insurance carriers like GEICO, State Farm or Allstate.
Depending on the location of your new property, you may be required (or want) to buy flood or earthquake insurance. This is similar to homeowners' insurance in that it would cover property and possessions, but would extend the types of events covered to included floods and earthquakes. Your homeowners' insurance carrier should be able to offer flood and earthquake insurance in addition to your homeowners' insurance.