Don't understand the Fundamental Theorem of Calculus 
If $f$ is continuous on $[a, b]$ and defining 
  $$
F(x) = \int_a^x \! f(t) \, dt
$$ 
  for $x \in [a, b]$, then $F'(x) = f(x)$ for $x \in (a,b)$.

I don't understand what function the variable $t$ is for.  What is the definition of $f(t)$?  And is $F(x)$ a set?  Because it seems like $\int_a^x$ can have varying answers depending on what $x$ is.

Well, the other answers are very satisfactory and more than sufficient for dealing with the problem, but I'd like to expose another point of view...
I hate this notation.
You don't integrate $f(t)$ on $[a,b]$. You integrate a function $f:[a,b] \rightarrow \mathbb{R}$ on $[a,b]$. This is illustrated by the presence of the so-called "dummy variable" being not only useless and immaterial, but also confusing. We can re-write it simply by
$$\int_a^b f(t)dt=\int_a^b f$$
And (with Lebesgue's Theory in mind), we get the much more illuminating notation:
$$\int_{[a,b]}fd\mu$$
$[a,b]$ is the set we are integrating over, and $\mu$ is the measure we are considering, which we can simply omit from notation in most cases when dealing with calculus, since we take the lebesgue measure.... arriving at:
$$\int_{[a,b]}f$$
Now, we define, for $x \in [a,b]$, a function $F:[a,b] \rightarrow \mathbb{R}$ by:
$$F(x)=\int_{[a,x]}f$$
Note that we are changing the set we are integrating over when we change $x$. The fundamental theorem of calculus is simply stating that if $f$ is continuous at $x$, then:
$$F'(x)=f(x)$$
Note: An "argument" for the notation is that it makes the theorem of change of variables intuitive. I disagree. It makes the theorem of change of variables (arguably) more easily manageable and prone to good memorizing and calculation, but it is as far as it goes. The intuition is independent of notation. Even with this "advantage", I find the theorem much less confusing theoretically when stated like this:
$$ \int_{[\phi(a), \phi(b)]} f=\int_{[a,b]} f\circ \phi . \phi '$$
