What is the meaning of $\frac{0}{0}$? I asked my teacher what is the real meaning of $\cfrac{0}{0}$, and the answer I got was "nobody knows".
I can't leave this subject "as is". I need a decent explanation, at least an explanation to why "nobody knows". I'm sure you'll come up with a few good answers.
 A: Math is logic.
$12/3=4$ because if you have twelve oranges and three kids, each gets four.
However, dividing by zero is illogical; what does it mean? It is no longer a math question, but a philosophical one.
Everything you do in math needs to make sense!
A: Well, if $0/0$ is well-defined then it should play nicely with our definition of division. Specifically, $a/b = c$ if $a = cb$. So there's gotta be some c such that $0c = 0$. Let's try 1. $0\cdot 1 = 0$, so $0/0 = 1$. 
Not too bad. Now let's try two. $0\cdot 2 = 0$, so $0/0 = 2$. Since equality is transitive, $1=2$.
Aaaaaand we just broke math. We resolve this by saying that $0/0$ is not defined, which keeps everything running nicely with a minimum of fuss.
A: I really hope your calculus teacher didn't say "nobody knows": $\frac{0}{0}$ is pretty well understood. It's just that it's undefined (without further context).
What I mean by "context" is: it's possible to give meaning to limits of the form $$\lim_{x \to a} \frac{f(x)}{g(x)}$$
where $\lim_{x \to a}f(x) = 0$ and $\lim_{x \to a} g(x) = 0$, but this depends on your choice of $f$ and $g$. For instance
$$\lim_{x \to 0} \dfrac{\sin x}{x} = 1, \qquad \lim_{x \to 0} \dfrac{e^{kx}-1}{x} = k, \quad \lim_{x \to 0} \dfrac{x}{x^3} = \infty, \quad \cdots$$
These are all different, despite the fact they're all of the form $\frac{0}{0}$.
So it's not that nobody knows, it's just that it's meaningless! But we can assign meaning if we know how the $0$s in the fraction came to be.
A: The expression $0/0$ is meaningless because the operation of division $(a,b) \mapsto a/b$ is not defined for pairs $(a,b)$ where $b$ is zero, just as it is not defined for pairs $(a,b)$ where $b$ is an elephant.
Sometimes you will hear that $0/0$ is an "indeterminate form" that can equal different things depending on the context.  This is a horribly imprecise way of speaking.  What it really means is that if you are evaluating a limit and you do the computation
$$\lim_{x \to 0} \frac{f(x)}{g(x)} = \frac{\lim_{x \to 0}  f(x)}{\lim_{x \to 0}  g(x)} = \frac{0}{0}$$
Then you have made a mistake in the first step and you have to evaluate the limit in a different way, e.g. with l'Hospital's rule, to get a valid answer (which could be anything depending on the particulars of the problem, hence "indeterminate.")  This reason that the first step in the displayed calculation is a mistake is that the rule "the limit of a quotient is the quotient of the limits" is not true in general—it is only true when the limit of the denominator is nonzero.
