What is this lower number? I was taught that the lower number in math would be the base, but you can't have base 0 (can you?)
I'm looking at some derivatives and it looks something like this.
$$x^2_0$$
Sorry for the stupid question, just trying to teach myself
 A: In your expression $x^2_0$ the "0" is called a "subscript", and the "1" is called a "superscript". Subscripts and superscripts have many different meanings in different contexts. 
Subscripts are sometimes used just to write a sequence of mathematical objects, such as a sequence of numbers: $x_1, x_2, x_3, x_4, \ldots$. But they are not always used that way.
Superscripts are sometimes used to write powers: $x^3 = x \cdot x \cdot x$. But they are not always used that way.
Without any context, it's impossible to tell you what $x^2_0$ means in whatever it is you are reading.
A: Subscripts are often used to create new variable names. After all, if you were only using single lowercase Latin letters, you could never refer to more than 26 different variables. 
Using subscripts allows you to create any number of new variables. Just as "$x$" and "$y$" are different variables, so are "$x_0$", "$x_1$", "$x_2$", etc., and these are all different from "$x$" itself.
There are, as mentioned in other answers/comments, lots of other uses for subscripts and superscripts, but this use is quite common and is likely to be what you are seeing.
It is also traditional to use subscripted variables such as "$x_0$" to stand for some constant (but not yet specified) value of "$x$".
For example, the equation of a circle of radius $1$ centered at the point $(x_0,y_0)$ is
$$(x-x_0)^2 + (y-y_0)^2 = 1$$
so that, for example, the unit circle centered at the origin is recovered by taking $x_0=0$ and $y_0=0$. And notice that actually expanding the left side of the equation above yields an equation containing the very term you asked about:
$$x^2 - 2x_0x + \boxed{x_0^2} + y^2 - 2y_0y + y_0^2 = 1.
$$
