What exactly do these integral functions mean? $$F(x) = \int{f(x)}\,dx$$
$$G(x) = \int_0^x{g(z)}\,dz$$
I am confused about the exact meaning about these functions.  The second function is clear to me, $G(x)$ is just the area under the graph of $g(x)$ from $0$ to some $x$.  But the first function is not so clear.
Also, why is the following considered incorrect?
$$H(x) = \int_0^x{g(x)}\,dx$$
 A: These are good questions.
The notation $\displaystyle \int f(x) dx$ is shorthand for "an antiderivative of $f(x)$." That is, a function with the property that $F'(x) = f(x)$. Part of the depth of the fundamental theorem of calculus is that antiderivatives are also ways to calculate the area under a curve. This is remarkable - why should they be related?
The last piece you wrote, $H(x) = \displaystyle \int_0^x g(x) dx$ is terrible notation, and should not be written in front of a non-expert crowd (experts can take in notational abuse much easier than neophytes). It it far more correct to write
$$
H(x) = \int_0^x g(t) dt
$$
to not confuse dummy variables and the actual variable.
A: $$
H(x) = \int_0^x{g(x)}\,dx
$$
is not incorrect (just confusing for unexperienced students).  This function $H$ is an antiderivative of $g$; that is $H'=g$.  But usually we write $G'=g$ and $F'=f$ since it is easier to remember what is what.  
Formula
$$
F(x) = \int{f(x)}\,dx
$$
is known as an indefinite integral and means $F'=f$.
