Calculus I is the usual title of the course that you take in a college or university. Calculus AB is the calculus that you take in high school for AP (advanced placent) credit for college, but it does not get you the same number of credits as a full college or university calculus course would get you. Calculus BC is another calculus that you would take in high school for AP credit, but this calculus gives you the same number of credits as had you taken it college. BC is simply a longer, more extensive course for the real high school math geniuses who can do it. AB calc is for the people who just want to get calculus out of the way and don't care about the credit as much.
Loosely speaking, Calculus AB can be thought of as equivalent to a university calculus course for life science majors. Calculus BC would be more equivalent to a university calculus course for engineering, physics, or math majors. Again, very loosely speaking.
