The simplest surfaces are those given by holding one of the coordinates constant. Thus, the $xy$-plane is given by $z=0$. Its (surface) area element is $dA=(dx)(dy)=(dr)(r\,d\phi)$, as can easily be seen by drawing the appropriate small rectangle. The surface of a cylinder is nearly as easy, as it is given by $r=a$ in cylindrical coordinates, and drawing a small “rectangle” yields for the surface element \begin{eqnarray} \hbox{cylinder:} \qquad && \dS = (a\,d\phi)(dz) = a\, d\phi \, dz \quad\qquad\qquad\nonumber\end{eqnarray} while a similar construction for the sphere given by $r=a$ in spherical coordinates yields \begin{eqnarray}\nonumber \hbox{sphere:} \qquad && \dS = (a\,d\theta)(a\sin\theta\,d\phi) = a^2 \sin\theta \, d\theta \, d\phi \end{eqnarray} The last expression can of course be used to compute the surface area of a sphere, which is  1) \begin{equation} \Int_{\rm sphere} \!\! \dS = \int_0^{2\pi} \int_0^\pi a^2 \sin\theta \, d\theta \, d\phi = 4 \pi a^2 \end{equation}

What about more complicated surfaces?

The basic building block for surface integrals is the infinitesimal area $\dS$, obtained by chopping up the surface into small pieces. If the pieces are small parallelograms, then the area can be determined by taking the cross product of the sides!