Having surgery with local anaesthesia means you will have a numbing medicine injected where the surgery will be, making that area numb so you will feel no pain during surgery.
Further information
Local anaesthesia is when a medicine called local anaesthetic is injected to make a small area of your body numb. It is a good option for some minor surgeries. You stay awake but should not feel pain. It is used by many doctors such as your GP for taking biopsies or your dentist for working on your teeth. In hospital it can be given by your surgeon for minor surgery. The numb feeling usually only lasts for a few hours.
It can also be given together with sedation.