Create a table with two parts with different tabular features

Within a table environment, you can use different tabular environments, of different types and with a different number of columns. Here's an example with sub captions:

\documentclass{article}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{tabularx}
\begin{document}
\begin{table}
\centering
\caption{An interesting table}
\subcaption*{Panel A: Some stuff}
\begin{tabular}{lcr}
First name & Last name  & Product \\
Bubba & Gump & Shrimp \\
Steve & Jobs & Happiness
\end{tabular}
\bigskip
\subcaption*{Panel B: Other stuff}
\begin{tabular}{ll}
School & State \\
Harvard & MA \\
Yale & CT \\
Brown & RI
\end{tabular}
\end{table}
\end{document}

Here I used the subcaption packages. A good alternative is the subfig package. However, the subfigure package is obsolete.

With liberal use of the \multicolumn{.}{.}{...} command, you can get away with spreading the table across the entire \linewidth:

\documentclass{article}

\usepackage{array,booktabs}

\begin{document}

\begin{table}[ht]
  \centering
  \caption{An interesting table}
  \label{tbl:interesting}

  \begin{tabular}{*{6}{p{.16\linewidth}}}
    \multicolumn{6}{c}{Panel A: Some stuff} \\
    \toprule
    \multicolumn{2}{p{.33\linewidth}}{First name} & \multicolumn{2}{p{.33\linewidth}}{Last name} & 
      \multicolumn{2}{p{.33\linewidth}}{Product} \\
    \midrule
    \multicolumn{2}{l}{Bubba} & \multicolumn{2}{l}{Gump} & \multicolumn{2}{l}{Shrimp} \\
    \multicolumn{2}{l}{Steve} & \multicolumn{2}{l}{Jobs} & \multicolumn{2}{l}{Happiness} \\
    \bottomrule
    \\
    \multicolumn{6}{c}{Panel B: Other stuff} \\
    \toprule
    \multicolumn{3}{p{.49\linewidth}}{School} & \multicolumn{3}{p{.49\linewidth}}{State} \\
    \midrule
    \multicolumn{3}{l}{Harvard} & \multicolumn{3}{l}{MA} \\
    \multicolumn{3}{l}{Yale} & \multicolumn{3}{l}{CT} \\ 
    \multicolumn{3}{l}{Brown} & \multicolumn{3}{l}{RI} \\
    \bottomrule
  \end{tabular}
\end{table}
\end{document}

Since the two panels are contained in the same tabular, they span the same width. The above uses the booktabs package for presentation of the tabular environments. However, it is not necessarily needed. If you want to drop it, you should also drop/replace the \toprule, \midrule and \bottomrule rules with \hline or another preference.

Alternatively, you could also use the tabularx package to spread columns across a specific width:

\documentclass{article}

\usepackage{booktabs,tabularx}

\begin{document}
\begin{table}[ht]
  \centering
  \caption{An interesting table}
  \label{tbl:interesting}

  \medskip

  \begin{tabularx}{\linewidth}{ X X X }
    \multicolumn{3}{c}{Panel A: Some stuff} \\
    \toprule
    First name & Last name & Product \\
    \midrule
    Bubba & Gump & Shrimp \\
    Steve & Jobs & Happiness \\
    \bottomrule
  \end{tabularx}

  \bigskip

  \begin{tabularx}{\linewidth}{ X X }
    \multicolumn{2}{c}{Panel B: Other stuff} \\
    \toprule
    School & State \\
    \midrule
    Harvard & MA \\
    Yale & CT \\ 
    Brown & RI \\
    \bottomrule
  \end{tabularx}
\end{table}
\end{document}

You can use the multicol column package to have data span multiple columns.

\documentclass{article}
\usepackage{multicol}

\begin{document}
\begin{tabular}{lll}
\multicolumn{3}{c}{Panel A: Some stuff}\\
First name      &Last name       &Product\\
Bubba           &Gump            &Shrimp\\
Steve           &Jobs            &Happiness\\
\\
\multicolumn{3}{c}{Panel B: Other stuff}\\
School       &State\\
Harvard      &MA\\
Yale         &CT\\
Brown        &RI\\
\end{tabular}
\end{document}