How to center cell contents of a LaTeX table whose columns have fixed widths?

\usepackage{array} in the preamble

then this:

\begin{tabular}{| >{\centering\arraybackslash}m{1in} | >{\centering\arraybackslash}m{1in} |}

note that the “m” for fixed with column is provided by the array package, and will give you vertical centering (if you don’t want this just go back to “p”