As a disclaimer, this is absolutely not a knock against any veterinary general practitioner! For many, many years the GP would do (and still does!) many surgeries and procedures a human GP would never dream of doing. Can you imagine a human GP doing a major abdominal surgery like a hysterectomy (spay)? However, some issues are likely best seen by a specialist and a good general practitioner knows when to refer. While specialists are usually more expensive to go see than a general practitioner, it can be very valuable for the livelihood of your working or performance dog to not delay in seeing a specialist given the time and money you have put into your dog.
In veterinary medicine, to call yourself a specialist, you must attend undergraduate college for four years and four years of veterinary medical school. After vet school, most candidates complete one or more years in a rotating internship where they see additional cases in a broad variety of specialties typically either at an academic teaching hospital or a larger private practice. After the internship, a residency in a specialty usually between two and four years depending on the specialty is completed and candidates sit for a board exam. Then they can be called a specialist once they pass the strenuous board exam.
Specialties in veterinary medicine are just about as broad as in human medicine and can range from anesthesiology, dentistry, internal medicine, and radiology to neurosurgery, radiation oncology, and zoo medicine and surgery. There are general practitioners with special interests in certain areas (for instance, my special interests are in nutrition, sports medicine, and reproduction) that go to continuing education to further hone their skills, but these cannot be called "specialists."
So how does one find a specialist for their dog? Ideally, your local vet can help them locate and refer you to a specialist. This is helpful as your local vet will have seen your dog progressively for the issue and can relay them to the specialist along with sending over necessary medical history, radiographs (x-rays), and so on. But if you're not sure where to turn, here are a few common "colleges" (kind of like the overseeing governing boards for a certain specialty) where you can look up local specialties.
American College of Veterinary Internal Medicine (includes internal medicine, neurology, cardiology, oncology)
American College of Veterinary Surgeons
American College of Veterinary Sports Medicine and Rehabilitation
American College of Veterinary Nutrition
American College of Ophthalmology
American Veterinary Dental College:
The preceding list is not exhaustive and there are many other specialties out there. In addition to the specific specialty colleges, the nearest veterinary teaching hospital is also a good place to inquire.