The MBA is a general degree preparing students for management positions in any level of a business, up to CEO. MBA students study accounting, finance, marketing, management, statistics, economics, strategy, policy, and other courses. Many MBA programs offer concentrations in these and many other fields, but that amounts to only 2-3 courses in your chosen field in the second year of study. Many students avoid a concentration and take a variety of elective subjects to gain a broader background. You don't become a specialist in a field with 2-3 courses. It generally takes a year of concentrated study.
The MS degree concentrates study in a single field and prepares students for high level staff or research positions. The MS typically requires an undergraduate education in the field in which you want the MS, or a closely related field. A finance major does not get an MS in chemistry, and a biology major does not get an MS in accounting. If you want to specialize in a particular field other than business administration, such as finance, marketing, operations management, human resource management, or a non-business field such as public health, or public administration, you should get an MS degree in that specialization.
MBA programs accept students in any undergraduate field. They prefer students who do not have a business background because they give you the business training but they cannot provide the broad background that managers should have. I have taught MBA students with degrees in Music, Medicine, Dentistry, Law, Psychology, Political Science, Chemistry, Biology, Engineering, and many other fields. Most MBA programs prefer students with 2-3 years work experience after the first degree, but many accept students right out of college if they have good grades and a high GMAT score. Some MBA programs are designed specifically for new college graduates without work experience. Typically students can go into MS programs right out of their undergraduate program.
The Official MBA Guide provides a lot of information about the MBA degree and MBA schools and programs. It's a comprehensive free public service with more than 2,000 MBA programs listed worldwide. It allows you to search for programs by location (US, Europe, Far East, etc.), by concentration (finance, marketing, aviation management, health management, accounting, etc.), by type of program (full-time, distance learning, part-time, executive, and accelerated), and by listing your own criteria and preferences to get a list of universities that satisfy your needs. Schools report their accreditation status, tuition cost, number of students, class sizes, program length, and a lot of other data. Schools provide data on entrance requirements, program costs, program characteristics, joint degrees, and much more. You can use the Guide to contact schools of your choice, examine their data, visit their web site, and send them pre applications. You can see lists of top 40 schools ranked by starting salaries of graduates, GMAT scores, and other criteria. It's the best service available at http://officialmbaguide.org.