Graduate School? 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.
 A: 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. 
A: I can only give you a definitve answer to 1), 2) and 3) seem to be dependent on the country/university/college/etc. that you are visiting. Since I'm from Europe, I cannot
give you complete answers for 2) and 3)...
Now to the answers:
1) You will receive a master's degree when you successfully complete graduate school. For this, you usually study for about 1.5 - 2 years (after you finished undergraduate school...) and write a master thesis, as well as do the final master exams.
To get a Ph.D., you need to have visited and completed graduate school, then find a research group which will fund your Ph.D. thesis. This thesis is an autonomous research project, which usually takes about 3 to 4 years to finish. At the end, your thesis will be evaluated and you need to pass an exam about your project (or rather, the subject of your project). Granted you pass, you will receive a Ph.D., which is roughly speaking, the first "real" academic title you can get. Afterwards, you can change into the private sector, where you will generally assume some leading position in research or administration, or you stay at the academy and continue towards your professorship. 
[tl;dr]: M.Sc. $<$ Ph.D. (regarding ability, know-how, experience and future job possibilities)
2) Here I cannot really give you an answer, since I do not study in the USA. But clearly you will need to complete undergraduate school. Which qualifications you will nedd to attend graduate school in mathematics, I cannot tell...
Yet doing some research projects on the undergraduate level can never be wrong, as long as you still have enough time to complete what you need to complete. It is especially useful for gaining insight into your prospective field or method of research, which might reveal that something else would suit you better, or that you are exactly where you need to be.
3) Basically, as with your other questions, you should do a consultation on your future career. I'm sure your undergraduate school will have some sort of advisory service. They will most certainly know exactly what you'll need and where you ought to go. Last but not least: Take your time choosing your field of study! It can be a quite painful quest, and you should still be able to change fields of study/research later on (as long as they are at least somewhat connected, so with math you would have a good starting point to get into physics, chemistry, biology, even things like psychology, medicine, etc.)
I hope that clears out some of your questions. Also, you might want to consider editing your question, since you may not give enough information (your school for a start...) to get satisfying answers.
Anyway, I hope I could help!
Sincerely,
SDV
A: From what I understand you want to continue your studies in mathematics. You have realized that one can so a Master degree and a Ph.D. degree, but you want to know what the difference is and what it requires. You will get different answers from different people because people have difference backgrounds and different philosophies. People do mathematics for different reasons. So the following is just my take on all of this.
While some of the answers might depend on your geographic location, some of the general ideas should be the same all over.


*

*The first thing I would strongly recommend (I mean strongly recommend) is that you talk to the faculty at Stony Brook. On the department website you have the contact information for the Graduate Director. This would be a very good person to talk to. This person should be able to answer all your questions about the concrete graduate program at Stony Brook. But I would also try to approach other faculty and ask them about what research they do and what research is like. You could try to talk to one of the professors that you currently are taking a class with. A person like that might be able to say whether or nor you would fit well with the graduate program at Stony Brook. I know that it can be intimidating to approach a professor, but (in my experience) most really like to see an interested student. Besides, for most professors it is part of their job description that they interact with students like you. The second thing that I can't recommend enough is to **befriend some of the (other) graduate student**s in your department. Again, this might seem awkward, but they are the ones who are living the life that you are interested in. Do the graduate students have social events that you are allowed to attend?

*The second thing that you should do is to give some thought to what you would like to do in life in general. Could you see yourself as a researcher in mathematics? Would you like to teach? Do you absolutely never want to teach? Would you like to work in the industry? Do you have other interests that might overlap well with mathematics? Does working for the government (for exmaple NSA) sound appealing? 



*

*So here is the thing. A Ph.D. is a program that trains you and prepares you for doing research in mathematics. That means high level research. Most researchers will also be required to do some teaching besides their research. So if you hate teaching, then you might not want to aim for a research career. Does that mean that you have to do research with a Ph.D.? No, differently not! Some people get a Ph.D. and end up teaching full time at maybe a smaller college where there is not research requirement. Some Ph.D.s go into the industry. Some, for example, get jobs on Wall Street! But this doesn't change the general philosophy that a Ph.D. is an education in how to do research. You will (most likely) be required to do some independent research for your Ph.D. in order to graduate. A Master degree can also involve research. Many in the US take the Master degree (MS or MA) before they even decide whether or not they want to continue with the Ph.D. Some start the Ph.D. but decide to stop when they are awarded a Master degree. A graduate director/advisor should be able to tell you how exactly this works for Stony Brook. That said, a MS degree usually is aimed more industry. 

*How do you know what "you need" to go to graduate school? Again, the graduate director will be able to give you the details about what is required. If you are thinking about other schools, you might find the answer to this question by visiting the websites of the departments. But there are, of course, more to graduate school than meeting some formal requirements. For the Ph.D. I would personally say that the key requirement is a love of mathematics. A Ph.D. can take years to complete and it will (for most) seem hopeless at times. Here your deep love of mathematics and learning is what I believe should/will carry you through. If you are supposed to do mathematics as an independent researcher, then you have to be willing to study for not other reason that you enjoy it. That said, it also requires some dedication. You have to invest many hours of hard work.

*How do you know what to study? I would again say that you should talk to your professors and ask them what they do. That way you can get an insight into how research is done and how researchers work with different topics. You also will get a feel for different personalities which will help you in possibly choosing an advisor at some point. If you like differential equations, then maybe look at the department website for people who work in that area and contact them. It looks like there is a whole group working with partial differential equations. You might also consider applied mathematics and mathematical physics. I will also say that some start their Ph.D. without knowing what they are interested in. They know that they like mathematics. But they are not sure what area they like the best. If you are in that situation, I might just recommend that you find a professor whom you like and whom you could see yourself spending time with. Here a good advisor could be someone who, besides being smart in their area, also is approachable. You want someone that has time for you and is willing to help when you are stuck. I recommend also looking up the professors online to see if they are actively publishing. I think it is best to have an advisor who is actively engaged in research since that increases the prospects of you getting a good result.
