Many U.S. universities make a distinction between “graduate” programs and “professional” programs.
A graduate program generally is one that leads to a degree of M.A., M.S., or Ph.D. They are considered academic programs, and not programs leading to a professional qualification. They usually (always for Ph.D.) require a thesis/dissertation.
Professional programs, on the other hand, usually lead to a specific degree in a specific area, such as M.Ed., M.B.A., M.S.W., J.D., M.D., D.D.S., etc. The programs are usually more structured than “graduate” programs, and often do not require a thesis.
A note on the J.D. degree. Traditionally, U.S. law schools awarded their graduates the LL.B. degree (bachelor of laws). In the 1950’s (I believe), some lawyers realized that the federal government paid its employees with doctoral degrees at a higher rate. They convinced their law schools to award a J.D. (juris doctor) degree rather than an LL.B. (and to offer to convert the degree for existing graduates), thereby qualifying for a higher pay scale.
This leads to a bit of a quirk in the advanced law degrees. The basic law degree awarded is a J.D. After your J.D., you can go for another law degree, an LL.M. (master of laws), which is usually a 1-2 year program in a specific area of law (commonly tax). The highest law degree is an S.J.D. (doctor of juridical science), which is an academic degree requiring a dissertation like a Ph.D. The degree of LL.B. (doctor of laws) is almost exclusively given out as an honorary degree in the U.S.
I understand that something similar is often the case with medical doctors in Commonwealth universities. Rather than the M.D. (medical doctor) degree universally awarded in the U.S., doctors will earn a M.B. (bachelor of medicne) and/or Ch.B. (bachelor of surgery) as their basic medical degree.
On the undergraduate level, the difference between a B.A., B.S., and other bachelors degrees depends heavily on the policies of the individual college/university. Often in a university, different subjects will be administratively grouped into different “schools” or “colleges,” which may award different degrees.
Where I went as an undergraduate, (University of Pennsylvania), there were four schools that awarded undergraduate degrees. The College of Arts and Sciences awarded the B.A. degree for all of its programs, including the hard sciences. The Nursing School awarded a B.S.N. (bachelor of science in nursing). The Wharton School (business school) awarded a B.S.Econ. (bachelor of science in economics) to its regular graduates, and a B.B.A. (bachelor of business administration) to its graduates from its evening school. The School of Engineering and Applied Science awarded two degrees, a B.S.E. (bachelor of science in engineering), which is the usual degree to qualify as a beginning engineer, and a B.A.S. (bachelor of applied science), which is a less rigorous, non-professional engineering degree.
Some other universities will offer B.A. degrees to arts and humanities majors and B.S. degrees to science majors. Others will offer both B.S. and B.A. degrees in the same fields, with the B.S. degree usually being more rigorous with more math/science requirements. Other common bachelors degrees include B.S.B.A. (bachelor of science in business administration), B.F.A. (bachelor of fine arts), B.Mus. (bachelor of music).
I have no idea how colleges choose which bachelors degrees to award for their programs, particularly when there are several that are essentially identical (B.B.A./B.S.B.A./B.S.Econ.)