Judy Reese and her service Dog, Shakespeare, helped us understand the difference between service dogs, therapy dogs and emotional support animals. Judy is an expert as she has had a service dog to warn her of impending seizures since she was 8 years old living on the family horse farm. 
Service dogs, or more properly service animals, were recognized in law through the Americans with Disabilities Act. The only service animals protected by the law are service dogs and miniature horses. They are specially trained animals who perform therapeutic tasks for their owners. The tasks they perform range widely: replacing or improving vision and hearing; providing medical alerts; allergy; mobility support; PTSD and other psychological support. Each is trained to serve one single person and his/her unique needs. The training takes years so the initial cost for a service dog is about $25,000, and there are long waiting lists. 
For example, Shakespeare, a Bouvier des Flanders, warns Judy of an impending seizure 15 to 30 minutes before it happens. This allows Judy to sit or lie down so she won't be injured through the inevitable fall the seizures induce. Bouviers were originally bred as cattle dogs, but have been service dogs for over 100 years. During World War 1 they were used by the military to find wounded soldiers and string communications wires across battle fields at night. They were effective because of their standard black coloring and high intelligence. 
Trained dogs provide service for about 10 years after formal training, which starts when they are puppies, before the aches and pains of age interfere with their capabilities. Miniature horses are able to provide service for 15 to 20 years, and yes the miniature horses are house trained just as the dogs are. 
Under the ADA, service animals are able to go anywhere with their masters. The only questions you can ask about a service animal are "Is it a service dog (or horse)?" and "What is the task it is trained for?" These are useful questions to ask since many people are ordering service animal vests, documentation, and  even medical scripts online in order to be able to take their pet with them everywhere. If the owner cannot or will not answer these questions, they are likely not truly service animals and you don't have to give them access.
A way to visually check if an animal is trained for service is to watch how it positions itself around the owner. Service dogs will essentially always put themselves between their owner and other people. They are trained to protect the owner, and this is a standard behavior reflecting this. 
Even if an animal is a true service animal you can ask the owner to remove them from your premises if they are disruptive in any way, including messing inside. The owner is required to properly care for their service animal and the animal has spent years in training to learn how to handle essentially any situation before starting service, therefore no disruptions are allowed. 
Therapy dogs have no special training, They are pleasant dogs who are taken to hospitals and other locations to help patients by being there with them. They provide real help to people but they are not true service dogs. 
Emotional support animals are essentially pets. Any animal qualifies and no training is required. The designation is used to override landlord restrictions on housing animals. Airlines recognized their use as a calming influence during flights. However, there is no definition in law unless it is in local laws regarding housing. Now that many people are trying to use these labels to get free flights for their pets, including birds, snakes and a wide variety of other animals. airlines are redefining their usage of the term. Judy, and many of us agreed, essentially all dogs are emotional support animals with the way the freely give us their love and attention.