Vertical spacing of a table cell

I have solved this in the past by modifying the value of \arraystretch

you can do this by adding to your source :

{\renewcommand{\arraystretch}{1.2} %<- modify value to suit your needs
\begin{tabular}{|c|c|}
...
\end{tabular}
}

Edit:

Actually having researched this a bit more it seems that having equations in the cells has unexpected behaviour, where the space is increase disproportionally at the top (value set to 3 to start getting the space at the bottom):

more recently I have started to use the tabu package to replace all tables (from tabular to tabularx to longtable). It also provide a few more controls. in this case the \tabulinesep has a much more appropirate effect on the results:

\documentclass[preview]{standalone}
   \usepackage{tabu}
   \begin{document}
   {\tabulinesep=1.2mm
   \begin{tabu} {|c|c|}
       \hline
       Cylindrical & $\displaystyle{{1 \over \rho}{\partial \over \partial\rho}\left(\rho {\partial f \over \partial \rho}\right)
       + {1 \over \rho^2}{\partial^2 f \over \partial \phi^2}  + {\partial^2 f \over \partial z^2}}$\\\hline
       Spherical & $\displaystyle{{1 \over r^2}{\partial \over \partial r}\!\left(r^2 {\partial f \over \partial r}\right)
       \!+\!{1 \over r^2\!\sin\theta}{\partial \over \partial \theta}\!\left(\sin\theta {\partial f \over \partial \theta}\right)
       \!+\!{1 \over r^2\!\sin^2\theta}{\partial^2 f \over \partial \phi^2}}$\\\hline
   \end{tabu}}
\end{document}

which produces the following result:

You may use a different table format, leaving out the vertical lines which are by no means necessary; you can also greatly simplify the input with a personal command for partial derivatives:

\documentclass{article}

\usepackage{booktabs,amsmath}

\newcommand{\dpder}[3][]{\dfrac{\partial^{#1}#2}{\partial #3}}

\begin{document}
\begin{tabular}{lc}
\toprule
Cartesian &
$\dpder[2]{f}{x^2}+\dpder[2]{f}{y^2}+\dpder[2]{f}{z^2}$
\\
\midrule
Cylindrical &
$\dfrac{1}{\rho} \dpder{}{\rho}{\left(\rho \dpder{f}{\rho}\right)}
  + \dfrac{1}{\rho^2} \dpder[2]{f}{\phi^2} + \dpder[2]{f}{z^2}$
\\
\midrule
Spherical &
$\dfrac{1}{r^2}\dpder{}{r}{\left(r^2 \dpder{f}{r}\right)}
  +\dfrac{1}{r^2\sin\theta}\dpder{}{\theta}{\left(\sin\theta \dpder{f}{\theta}\right)}
  +\dfrac{1}{r^2\sin^2\theta}\dpder[2]{f}{\phi^2}$
\\
\bottomrule
\end{tabular}
\end{document}

Notice how to avoid unwanted spaces in the "cylindrical" and "spherical" rows by using {\left(...\right)} (a couple of braces is sufficient instead of explicit backing up). Note also the usage of \dfrac in the definition of \dpder that avoids specifying \displaystyle (which is however not a command taking arguments, but a declaration).

With booktabs there's rarely the need to adjust the row spacing.