Betterment is a condition relating to original equipment, or older parts, on your automobile, when having been damaged in an accident may only be replaced with new part(s). The resulting improved condition of the vehicle is known as betterment. Consequently many insurers may ask the insured to contribute to the cost of the repair.
It will depend upon state laws and insurance company guidelines but if you want better parts or newer parts than what the insurance company is offering to pay for you may be able to receive them if you are willing to pay the difference.
So the basic definition of betterment is that your repaired vehicle is better than it was before it was damaged. Depending upon state laws it may be that your insurance company may only reduce your settlement if your vehicle's resale value has increased over what it was before the accident.
Generally, an insurer will deduct the difference between the cost of a used part (appropriate for age and condition of the vehicle) and the cost of a new part. This thus would be a betterment deduction.
For more information on this subject you can check with your insurance agent or your state's insurance regulator.