You had an additional spurious closing brace in the first C{2.6cm} column; delete it. I used a \makebox to center the table with respect to the text area:
\documentclass{article}
\usepackage{array}
\newcolumntype{L}[1]{>{\raggedright\arraybackslash}m{#1}}
\newcolumntype{C}[1]{>{\centering\arraybackslash}m{#1}}
\begin{document}
\begin{table}
\makebox[\linewidth][c]{\begin{tabular}{| L{4.2cm} | C{2.6cm} | C{2.6cm} | C{2.6cm} | C{2.61cm} |}
\hline
& \bfseries 1988 & \bfseries 1989 & \bfseries 1990 & \bfseries Change \\
\hline
Score & 249 & 234 & 266 & +17 \\
\hline
Percent High & 14 & 9 & 26 & +12 \\
\hline
\end{tabular}}
\end{table}
\end{document}
Some comments:
I changed the definitions of your columns, suppressing the \let and \hspace commands.
\bf is an old command that shouldn't be used anymore; use \bfseries instead; since cells form a group there's no need for explicit grouping.
As a suggestion, don't use the too restrictive option [!h] as placement specifier; use a less restrictive one (or don't use any at all).
Please (as a suggestion), consider using the booktabs package for your tables (they will look a lot better; no vertical rules, though).
Since your table will have numeric values, you could consider using the siunitx for possible alignment.
Just for comparison, the original table and the same table using booktabs:
\documentclass{article}
\usepackage{array}
\usepackage{booktabs}
\newcolumntype{L}[1]{>{\raggedright\arraybackslash}m{#1}}
\newcolumntype{C}[1]{>{\centering\arraybackslash}m{#1}}
\begin{document}
\begin{table}
\makebox[\linewidth][c]{\begin{tabular}{| L{4.2cm} | C{2.6cm} | C{2.6cm} | C{2.6cm} | C{2.61cm} |}
\hline
& \bfseries 1988 & \bfseries 1989 & \bfseries 1990 & \bfseries Change \\
\hline
Score & 249 & 234 & 266 & +17 \\
\hline
Percent High & 14 & 9 & 26 & +12 \\
\hline
\end{tabular}}
\end{table}
\begin{table}[!ht]% [!ht] used just for the example
\makebox[\linewidth][c]{\begin{tabular}{ L{4.2cm} C{2.6cm} C{2.6cm} C{2.6cm} C{2.61cm}}
\toprule
& \bfseries 1988 & \bfseries 1989 & \bfseries 1990 & \bfseries Change \\
\midrule
Score & 249 & 234 & 266 & +17 \\
Percent High & 14 & 9 & 26 & +12 \\
\bottomrule
\end{tabular}}
\end{table}
\end{document}

\hline not alignederror in LaTeX, which made your question initially very confusing. The error you got is! Misplaced \noalign. \hline ->\noalign– David Carlisle Sep 19 '13 at 08:23