A [b]service dog[/b] is individually trained to perform tasks that mitigate the disability of his owner. Training typically takes 18-24 months. Because of his advanced training, a service dog is considered medical equipment and is permitted to accompany his disabled owner to many places where pets are not permitted.
An [b]emotional support animal[/b] belongs to a person who is disabled. The person's doctor has determined that the presence of the animal is necessary for the disabled person's mental health and written a prescription stating the pet is necessary in the person's home, despite any "no pets" regulation of the landlord, for the person's health. Little or no training is required. The owner of an emotional support animal has no more right than any other pet owner to take their emotional support animal with them other to keep one in a home where pets are not permitted or to fly with one in a cabin when pets are not permitted.
A [b]therapy dog[/b] is a pet that has been trained, tested, registered, and insured to accompany his owner to visit patients and residents of facilities like hospitals and nursing homes to cheer up the people living there. A well-behaved pet can typically complete training in about 8 weeks. A therapy dog is legally a pet. It is not permitted to go anywhere that pets aren't without permission from the facility owner. The objective of registration is to show facility managers that this dog is well behaved, safe around people, and insured against liability. It is not a license to walk into a hospital or nursing home without permission.
In short: service dog works to help the owner perform tasks he cannot perform on his own because of his disability, an emotional support animal works to improve the health of his owner who is disabled, and the therapy animal works with his owner to improve the health of others.