"Local" = "local anesthetics", like lidocaine, bupivacaine, etc. These medications block nerve conduction, and therefore block sensation, pain, and motor function
If a surgery is scheduled as a "local", in general that means the surgeon/proceduralist is injecting the local anesthetic. No anesthesia personnel are involved in the procedure, so there is no additional sedation. Examples of procedures commonly performed under local alone are dental procedures, temporal artery biopsies, some toe and finger procedures, etc
If a surgery is scheduled as a "peripheral nerve block," then the anesthesiologist injects local anesthetics around specific nerves that cover the region in which an operation is taking place. They do this before surgery. Anesthesiologists are involved the entire procedure, and most commonly give IV sedation as well to make the patients more comfortable. Examples of surgeries commonly performed this way are hand and arm surgery, foot surgery, sometimes shoulder surgery, etc.
However to confuse things, anesthesiologists also commonly refer to "local anesthetics" as just "local". So they may tell a patient "I will inject local." Also, surgeons may tell patients they are "just having local", when they are really referring to the patient receiving care from an anesthesiologist, and just not needing a general anesthetic.
So the terminology depends on the context and the caregiver.