As the name implies, the Fundamental Theorem of
Calculus (FTC) is among the biggest ideas of
Calculus, tying together derivatives and
integrals. One half of the theorem gives the
easiest way to compute definite integrals. The
other half relates the rate at which an integral
is growing to the function being integrated. It is
the 5th of the
Six
Pillars of Calculus:

The whole change is the sum of the partial changes.

The Fundamental Theorem of Calculus relates three very different
concepts:

The definite integral $\int_a^b f(x)\, dx$ is the limit
of a sum.
$$\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x,$$
where $\Delta x = (b-a)/n$ and $x_i^*$ is an arbitrary point somewhere
between $x_{i-1}=a + (i-1)\Delta x$ and $x_i = a + i \Delta x$.
The name we give to the variable of integration doesn't matter:
$$\int_a^b f(x) \,dx = \int_a^b f(s)\, ds = \int_a^b f(t)\, dt$$

The indefinite integral is the function $$I(x) = \int_a^x f(s)\,
ds.$$ That is, it is a running total of the amount of stuff
that $f$ represents, between
$a$ and $x$. If $f$ is the height of a curve, then $I(x)$ is the area
under the curve between $a$ and $x$. If $f$ is velocity, then $I(x)$
is the distance traveled between time $a$ and time $x$.

An antiderivative $F(x)$ of $f(x)$ is a function with
$F'(x)=f(x)$. There are actually many different anti-derivatives of
$f(x)$, but they differ by constants. For instance, $x^3$ and $x^3+7$
are both anti-derivatives of $3x^2$.

When studying the Fundamental Theorem of Calculus, it's very important to keep
these straight!