When one talks of blood 'types', one is actually referring to the antigens on the red blood cell surface. (An antigen is a molecule, or part of one, that the immune system can target.)
There are two systems of antigens that typically cause the biggest immune reaction - the ABO group and the Rhesus group. Most blood types are reported only with reference to the two major systems. Thus, for the ABO group, you can be A, B, AB, or O; and for the Rhesus group, you can be positive or negative. (E.g. A+)
However, just because those two groups cause the biggest reaction (and thus are the most dangerous in a blood transfusion), it doesn't mean that these are the only types of antigens on the red blood cell surface. In fact, there are at least 30 commnly occuring antigens and hundreds of rare antigens. (Options like 'kell' and 'duffy' are among these.)
However most of these are weak and therefore of importance principally for studying the inheritance of genes to establish parentage.
Check out the following web site for a more detailed look at other blood types, including a list of the semi-common ones. http://www.bloodbook.com/type-sys.html...
One last foot note: the 'bombay' in your question is not actually an alternative to the ABO system. Rather, the Bombay phenotype is a weird quirk of that system. If you're interested, you can go here: http://www.madsci.org/posts/archives/Jan2003/1044040015.Ge.r.html