You are a vet if you served in the Armed Forces, or if you went to veterinary school and now give medical care to animals.
The noun vet is short for either veteran (of the Armed Forces) or veterinarian (animal doctor). As a verb, vet means "to carefully examine beforehand." For example, your boss might ask you to vet people's applications before calling them for interviews, just to make sure they have the right experience for the job. Medically speaking, when a doctor vets you — or an animal — medical care is given.