 $\displaystyle{\iint_R f(x,y) \, dA}$ is the double
integral of $f(x,y)$ over the region $R$.
 $\displaystyle{\iint_R f(x,y) \, dx\,
dy}=\displaystyle{\iint_R f(x,y) \, dy \,
dx}=\displaystyle{\iint_R f(x,y) \, dA}$.
You can see this by noticing the area of the
small rectangle is $\Delta A = \Delta x \,\Delta y =
\Delta y\, \Delta x$, so the infinitesimal area is $dA =
dx\, dy = dy\, dx$.
 $\displaystyle{\int_a^b \int_c^d f(x,y)\, dy \, dx}$
is an iterated integral. We first treat $x$ as
a constant and integrate $f(x,y) \,dy$ from $y=c$ to
$y=d$. Call the result $g(x)$. It is a
function of $x$ and describes the volume of the column
with that value of $x$. We then integrate
$g(x)\, dx$ to get the volume of all columns
from $x=a$ to $x=b$.
 $\displaystyle{\int_c^d \int_a^b f(x,y)\, dx \, dy}$
is an iterated integral. We first treat
$y$ as a constant and integrate $f(x,y) \,dx$ from
$x=a$ to $x=b$. Call the result $g(y)$. It
is a function of $y$ and describes the volume of the
row with that value of $y$. We then integrate
$g(y)\, dy$ to get the volume of all rows from
$y=c$ to $y=d$
 In an iterated integral, the order of $dx$ and $dy$
tells you which variable to integrate first. When
in doubt, draw parentheses: $$\int_a^b \int_c^d
f(x,y) \, dy\, dx = \int_a^b \left ( \int_c^d f(x,y)
\,dy \right )\, dx.$$
 Fubini's Theorem says that you can evaluate
double integrals by doing an iterated integral in
either order, but sometimes one order is a lot
simpler than the other.
