Table layout with tabularx (column widths: 50%|25%|25%)

The code below defines two new columntypes: b for 'big' and s for 'small'

I followed the details of section 4.3 of the documentation to create these.

\documentclass{article}
\usepackage{tabularx}

\newcolumntype{b}{X}
\newcolumntype{s}{>{\hsize=.5\hsize}X}

\begin{document}

\begin{table}[htbp]
    \centering
    %\begin{tabularx}{\textwidth}{| X | X | X |}
    \begin{tabularx}{\textwidth}{bss}
        \hline
        Alpha     & Beta     & Gamma     \\ \hline
        0         & 2        & 4         \\ \hline
        1         & 3        & 5         \\ \hline
    \end{tabularx}
\end{table}
\end{document}

If you'd like to center the heading, then you can use something like the following, which defines a \heading command to center its argument

\documentclass{article}
\usepackage{tabularx}

\newcolumntype{b}{X}
\newcolumntype{s}{>{\hsize=.5\hsize}X}
\newcommand{\heading}[1]{\multicolumn{1}{c}{#1}}

\begin{document}

\begin{table}[htbp]
    \centering
    %\begin{tabularx}{\textwidth}{| X | X | X |}
    \begin{tabularx}{\textwidth}{bss}
        \hline
        \heading{Alpha}     & \heading{Beta}     & \heading{Gamma}     \\ \hline
        0         & 2        & 4         \\ \hline
        1         & 3        & 5         \\ \hline
    \end{tabularx}
\end{table}
\end{document}

Knowing that each column has two "tab separators" and 4/3's of an \arrayrulewidth, you can subtract that from a regular p-column specifier:

\documentclass{article}
\usepackage{tabularx}% http://ctan.org/pkg/tabularx
\begin{document}

\noindent
\begin{tabularx}{\textwidth}{| X | X | X |}
  \hline
   Alpha     & Beta     & Gamma     \\ \hline
   0         & 2        & 4         \\ \hline
   1         & 3        & 5         \\ \hline
\end{tabularx}

\noindent
\begin{tabular}{
  |p{\dimexpr.5\linewidth-2\tabcolsep-1.3333\arrayrulewidth}% column 1
  |p{\dimexpr.25\linewidth-2\tabcolsep-1.3333\arrayrulewidth}% column 2
  |p{\dimexpr.25\linewidth-2\tabcolsep-1.3333\arrayrulewidth}|% column 3
  }
  \hline
  \centering Alpha     & \centering Beta     & \centering\arraybackslash Gamma     \\ \hline
  0         & 2        & 4         \\ \hline
  1         & 3        & 5         \\ \hline
\end{tabular}
\end{document}

Centering is obtained using \centering within the required cell (knowing that you're dealing with a p-column, this is possible). The modification of this alignment requires a small correction (via \arraybackslash) to re-establish the use of \\ in a tabular.

An alternative is to use the tabu environment. You can give "weights" (width coefficients) to X columns (X[2]):

\documentclass{article}
\usepackage{tabu}

\begin{document}

\begin{table}[htbp]
    \centering
    \begin{tabu} to \textwidth {| X[2] | X | X |}
        \hline
        \centering Alpha & \centering Beta & \centering Gamma \\ \hline
        0                & 2               & 4                \\ \hline
        1                & 3               & 5                \\ \hline
    \end{tabu}
\end{table}

\end{document}