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I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:
What exactly is calculus?
I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I heard it was used to calculate the area under a curve on a graph. If anyone can help me with this, I'd much appreciate it.
I was recently in a similar situation. After finishing precalculus at my high school, when I was 15 I started taking calculus at my local university and studying higher mathematics on my own (out of the book "Modern Algebra: An Introduction" by John Durbin, which in retrospect seems laughably basic but at the time blew my mind). Three years later, I can say without a doubt that it is the best decision I've ever made. I ended up learning mathematics through a combination of taking classes at university, talking with students/professors, reading textbooks, and using this site. I did have one major advantage over you though, as my parents are both professors (although neither of them math professors) which made it easier for me to get into classes. However, I know of other people doing the same thing without any connection to the university. Here are some things I would recommend based on my experience:
Get an introductory textbook for some relatively advanced subject, such as Calculus, Linear Algebra, or Abstract Algebra. Read reviews online before choosing one to find one that is both rigorous and easy enough for beginners. I'd recommend Spivak for Calculus (take this with a grain of salt though, as I never read it but have heard good things about it) or Durbin for Abstract Algebra. Make sure it comes with plenty of exercises, and DO THEM. If you don't know how to do a problem, or if you've done it correctly, ask here!
If you have a university nearby, take advantage of it. Email a professor teaching an upcoming introductory course and explain your situation to him/her, and ask if you can sit in on the class. You might even be able to enroll in classes as a non-degree-seeking student, if the university allows this (most do) and you can afford it (if it's a state school, the tuition for a single course might not be too bad). Don't be afraid to talk about math with professors. It can be intimidating, but remember, these people have dedicated their lives to math. Almost uniformly, they LOVE it. Half of the time I had to find a way to break off a conversation with a professor because they were so engrossed in the math at hand.
Find something specific you don't understand. It may be a theorem, a proof, a concept, or even an unsolved problem, so long as it fascinates you. Figure out what you need to know in order to understand it, and start down the rabbit hole. The experience of coming to understand something like this can be very rewarding in addition to teaching you a great deal of mathematics. I've had several of these over the past few years, most recently an unsolved problem known as the Triangular Billiards Conjecture which I'm studying right now.
If you have any questions about my experience, feel free to ask. Good luck!
I will not explain calculus. There are many websites that do it very well, probably much better than I can, so I'll leave it to them. I will address prerequisites for calculus with an emphasis on procedure rather than deep understanding. At some point, you would do well to revisit these concepts in a slow, measured pace, being very careful and filling in all the details. However, your position seems more of an "in over my head" thing so I'll try to address it from that perspective.
There are only two big concepts in calculus, the derivitive (verb: to differentiate) and the integral (verb: to integrate). Both of these are special uses of the extremely broad concept of a limit, but many introductory calculus classes only touch on limits in a very superficial way. Certainly to do a first physics course with calculus you will not need a deep understanding of limits, only an appreciation for why they let us get the results we want.
If you do not have a good grasp on trigonometry, you can still do/understand calculus but it will seem rather artificial. You should know the sine and cosine functions and it will help to know the other four that usually accompany them. You should know the unit circle and the special values on it. Some basic identities will make your life easier.
For limits, you can get by with an intuitive understanding of functions, and of real numbers. It is more helpful to have a good intuition for rational numbers, for example you should know that there are infinitely many rationals in between any two real numbers. You must understand the notion of the domain of a function. Experience with rational functions is extremely valuable. To do calculations with limits, you should be very comfortable with simplifying rational expressions, including domain issues and extraneous solutions. Again, you can do without rational functions, but you are much better equipped to understand the significance of limits if you can manipulate them without much trouble.
For derivatives, you will need to be familiar with operations of functions: addition, subtraction, multiplication, division. A special emphasis on composition of functions: for you this will probably be the most important prerequisite for solving physics problems. You should be familiar with but do not need to be extremely comfortable with implicitly defined curves: for example, the circle is not given in $y=f(x)$ form, but it is still well-defined. You should understand the domain issues that can arise when converting implicitly defined curves into function form.
For integrals, there are a lot of skills you could need, but I will try and keep it to a bare minimum. I would not try to understand the real definition of an integral (your resources may call it a Riemann integral), but it is absolutely essential that you understand the intuition behind it, and its link to limits.
Of course you must be familiar with finding area of basic shapes. You must be extremely comfortable with reading $\Sigma$ notation; if not in reading it directly, then at least to translating it into $+$ notation (you do not need to be able to write $\Sigma$ notation well). However, the most useful skills for cracking integrals are pattern-recognition and persistence; they can sometimes require quite a bit of creativity to solve.
There is another (shorter) list which I think is equally important for your situation: things which you are not expected to know, but are expected to pick up during a calculus class. These include deeply understanding inverse functions, familiarity with theorems of the form "If … then there exists …", high comfort with implicit curves, high comfort with recognizing compositions of functions [you will need to pick this one up], the significance of variables as distinct from numbers, skill at visualizing 3D space, distinction between functions and their values at points.
At some point while learning derivatives, you will come across the notion of related rates. Please learn this very carefully. Many students struggle a lot with this section — including me — but it is a very important use of calculus for physics. Perhaps it will not come up directly in your class, but if you know it well you will see it hiding just behind the things you discuss.
Best Answer
In a nutshell, Calculus (as seen in most basic undergraduate courses) is the study of change and behaviour of functions and sequences. The three main points are:
And obviously (and maybe especially), how these relate to one another; the crowning jewel of Calculus is probably the Fundamental Theorem of Calculus, which truly lives up to its name and was developed by none other than Leibniz and Newton.