What it means to be a Service Dog in the USA
Service dog or assistance dog laws vary around the world. Here in the United States there is one law that answers all of your questions regarding service animals and that is the American with Disabilities Act or the ADA. Now as the name suggests, the people covered under this law need to be legally disabled, either physically and/or mentally. An individual with a disability is defined by the ADA as a person who has a physical or mental impairment that substantially limits one or more major life activities, a person who has a history or record of such an impairment, or a person who is perceived by others as having such an impairment.
So, you have a disability, this doesn't mean you can just bring your pet dog or miniature horse (yes, miniature horses can be service animals) with you by default.
First, the disability must be mitigated by a task or job that is performed by the animal. Not all disabled persons will have the same need or same benefits from having a service animal. For example, someone with mobility/stability impairment may function just fine with a cane OR they may benefit from the flexibility/size provided by a large dog that is trained to help them maintain their balance. A person with Obsessive Compulsive Disorder may use a service dog to interrupt harmful/obsessive behaviors, i.e. a task that they could not complete on their own. Many individuals with service animals also use alternate methods to accommodate their disability in the event their working animal is unable to assist them for whatever reason. It is important to note that this does not make the need of their service any less valuable or needed. Essentially the disabled person must have a "gap" in their abilities that the dog or horse "bridges."
Second, the dog or miniature horse must be specifically trained to perform the task needed. For a guide dog this would be the act of learning how to guide a person. For a dog like Henry, who is my service dog shown above, this involves being trained to find a person when I'm down. Tasks can also be natural, however it is rare. A natural task could be the ability to sense an oncoming seizure or a change in blood pressure. Sometimes the changes and signals the dog is picking up on are impossible to pinpoint. How the dog alerts the handler to these events is often natural and then shaped (or trained into a specific response like a nose nudge or gentle paw to the leg). A service animal without a task is not legally a service animal and is not covered under US law.
Other training? Although the ADA does not specify what obedience training an animal must have in addition to its task work it is required that the animal be under the handlers control at all times as well as housebroken. If these conditions are not met, the animal may be denied entry or asked to leave.
"But I registered my dog as a service animal!" I am sorry to be the one to break this to you but within the United States there is no mandatory registry, despite what many websites are trying to portray. Websites that boast "Take your dog anywhere" are unfortunately there for profit and hold no legal weight. They handout certificates and ID cards with price tags ranging from $20-200 and up. I understand it can be confusing and frustrating, especially if you have spent your hard earned money on one of these sites. These registries are no more official than a hand drawn doodle stating "access required." The only way to have a service animal is to qualify under the ADA. It is important to note that some states offer a voluntary registry and some organizations do offer certificates to their trained dogs, however neither of these things are legally required.