Teaching, as a specific act, is a closest to an occupation for which a person would be paid to perform certain duties.
Education is sometimes a business, especially when seen in light of the recent growth of the for-profit university sector.
Education, also, more than "teaching" can be seen as a social service where citizens are given access to skills and training (carried out by teachers) that will help them to participate in the job market.
I to believe it is all of the above. Teaching is a business because teachers are employed by institutions that receive revenue in one form or another from the student body. It is also an occupation for the same reasons. Most importantly it is a huge social service that incorporates aspects of psychology, sociology, the arts, and public service.
Most social service workers are professionals, though. I would describe it as both an occupation and a social service, though not a business, for the reasons described above. Teachers are professionals who are paid (though not, in most cases, particularly well) for their labor and expertise. Yet they also perform a very important job in society. These are not mutually exclusive ideas.
Teaching is an occupation, in most cases. It's a job that involves a contractual committment, that you go to every day, and perform according to your administrator's requirements.
You can say that teaching is part of a business whent it's done "for profit," like a tutoring service.
On the other hand, you might call it a social service when somebody does it for free, as a charitable act.
From the point of view of the teacher, it is an occupation. It is what we do for a living. We typically do not have our own teaching firm so it is not a business. From the point of view of society it is a social service because it is an important service that is provided to members of society free of charge (except, of course, for the fact that we pay taxes).
I would have to say that it is all of the above. It is an occupation, but, as a teacher, I do not like thinking of it like that (given it is defined as "a regular activity performed for payment"). It is also a business. Teachers are in control of their classrooms (like bosses and employees). Lastly, it is a social service. Not only are teachers educating students, some act as social workers, psychologists, and police (not they they should or should have to).