If your cat Fluffy seems dazed after a visit to the vet, it’s probably because the vet had to tranquilize her. In other words, Fluffy was given a drug to calm her down or make her sleep.
When doctors tranquilize a patient, it's usually to calm or relax the person before a medical procedure or after some trauma. It's more common to hear the verb tranquilize in the context of animal medicine or encounters with wild animals. For example, if a bear wanders into a suburban neighborhood, officials might use a dart gun to tranquilize it; they can then safely capture and relocate the bear. The root word, tranquil, comes from the Latin tranquillus, meaning "quiet."