How can I make the following table in LaTeX?

Here is an option without using any packages:

\documentclass{article}
\begin{document}
\begin{tabular}{|*{6}{c|}}
\hline
& 1 & 276 & & \multicolumn{2}{c|}{letters} \\
\cline{2-3}\cline{5-6}
\raisebox{.5\normalbaselineskip}[0pt][0pt]{A} &
\multicolumn{2}{c|}{numbers} &
\raisebox{.5\normalbaselineskip}[0pt][0pt]{B} &
a b c &
A B C \\
\hline
\end{tabular}
\end{document}
Instead of moving the letters A and B "up" into position in the middle of the two rows, one can also move then "down":
\documentclass{article}
\begin{document}
\begin{tabular}{|*{6}{c|}}
\hline
\raisebox{-.5\normalbaselineskip}[0pt][0pt]{A} & 1 & 276 &
\raisebox{-.5\normalbaselineskip}[0pt][0pt]{B} & \multicolumn{2}{c|}{letters} \\
\cline{2-3}\cline{5-6}
& \multicolumn{2}{c|}{numbers} & & a b c & A B C \\
\hline
\end{tabular}
\end{document}
Alternatively, one can also use multirow's functionality via \multirow{2}{*}{<stuff>} (or \multirow{-2}{*}{<stuff>}) in a similar manner to the above solutions.
Here is one more possibility to do it without any package. And I also added the most typical (and mentioned by Werner) version with multirow, which can be found all over the place on TeX.SX.
% arara: pdflatex
\documentclass{article}
\usepackage{multirow} % just needed for the second solution
\begin{document}
\begin{tabular}{|c|@{}c@{}|c|@{}c@{}|}\hline
A &
\begin{tabular}{c|c}
1 & 276\\\hline
\multicolumn{2}{c}{numbers}
\end{tabular} &
B &
\begin{tabular}{c|c}
\multicolumn{2}{c}{letters}\\\hline
a b c & A B C\\
\end{tabular}\\\hline
\end{tabular}
\begin{tabular}{|*{6}{c|}}\hline
\multirow{2}{*}{A} & 1 & 276 & \multirow{2}{*}{B} & \multicolumn{2}{c|}{letters}\\\cline{2-3}\cline{5-6}
& \multicolumn{2}{c|}{numbers} & & a b c & A B C\\\hline
\end{tabular}
\end{document}
