That's just like a definite integral, which is the limit of a sum, only viewed as a function of its upper limit. However, the Fundamental Theorem of Calculus tells us that $F'(x) = f(x)$, so $F(x)$ is an antiderivative of $f(x)$.

All antiderivatives are the same, up to adding a constant, so most people use the terms indefinite integral and anti-derivative interchangably. Even the constants match up. Changing the value of $a$ in $\int_a^x f(t) dt$ changes $F(x)$ by a constant, so both anti-derivatives and integrals are only defined up to a constant.

From now on, the notation $\int f(x)\, dx$ will refer both to the anti-derivative and the indefinite integral, and we'll usually just call it "the integral of $f(x)$ (with respect to $x$)" for short.

Here are the basic techniques for finding anti-derivatives, a.k.a. integrals.

Checking whether $F(x)$ is an anti-derivative to $f(x)$ is easy — just compute $F'(x)$. If $F'(x) = f(x)$, then $F(x)$ is an anti-derivative, and any other antiderivative must be of the form $F(x) + C$.