TITLE: Graduate School?
QUESTION [7 upvotes]: I'm completely clueless on the process, but on track to graduate in two years, so I have a few questions about what I should do.
1) What's the difference between a Master's degree in Mathematics and a Ph.D in Mathematics? I have always wanted to teach mathematics at the college level, and so my family has always joked that I'd be in school forever to get a Ph.D. What is the difference between the two, and how do I go about choosing which is right for me? I don't really know the difference between undergraduate and graduate school, except that the latter is supposed to be much more difficult, much deeper, and is the starting point for research for many people.
2) What do I need to go to graduate school? At the moment, I have a 3.6 GPA, which I guess is alright, but I don't know if it's good enough or if there's anything else I need. I have awful test anxiety, and so my GPA in mathematics specifically is probably a lot worse than my overall GPA boosted by my minor and general electives. I am working on getting an undergraduate research opportunity to help put weight behind those numbers. Is this an appropriate things to do in undergraduate, and is it going to help me go further?
3) I have to come to realize just how expansive mathematics is through my time on MSE. There are tons and tons of fields of mathematics with all kinds of interesting problems. At the moment I'm really fascinated with differential equations because that's what Stony Brook researches at the graduate level, so their undergraduate course is like an introduction. If I decide I want to do that, how can I find other schools with similar focuses? If not, how can I figure out what I do want to study? It seems I'll never experience even the surface of some of these fields that are particularly deep.
Sorry for the long post, but thanks for any insight any of you can offer.

REPLY [7 votes]: Here are some brief comments on your questions.
1) For most master's programs you have to pay tuition yourself (or take loans). For Ph.D. programs in math you do not; the department covers your tuition in exchange for you doing some type of teaching and/or grading. This is true generally of graduate programs in math and the sciences, and it's a compelling non-academic reason to prefer a Ph.D. program if you can get admitted to one.
2) You should have strong grades in a range of upper-level math courses. It's the upper-level math grades that people look at. Nobody will care how you did in history or biology courses. Similarly, the GRE math subject test is important while the GRE general test is not (the level of math on the general test is a joke for math majors, at the level of the SAT). 
3) Don't feel committed to knowing what you want to work on in graduate school before you get there. That means two things: (1) If you think that you love differential equations and write in your application that this is what you want to do, nobody is going to hold you to that when you arrive in graduate school. Unlike lab sciences, where an incoming graduate student is attached to a particular professor's lab early on, in math graduate school you take courses and explore for a little bit, so if you switch interests nobody will hold it against you (it happens all the time, since there are many areas of math at the graduate level that you'd never see as an undergraduate). (2) If you don't want to commit in your application to having a definite area of interest, you don't need to, and could simply write about the areas you currently find interesting and why.