\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{multirow}
\title{Test}
\author{Chad Stucki}
\date{September 2016}
\begin{document}
\maketitle
\section{Introduction}
\begin{table}[!b]
\centering
\caption{Caption}
\begin{tabular}[t]{|c|c|}
\hline
\multicolumn{2}{|c|}{\rule{0pt}{2.5ex} \textbf{Class I Fairing}} \\\hline
\rule{0pt}{2.5ex} \textbf{Configuration} & \textbf{Drag Ratio} \\\hline
\rule{0pt}{2.5ex} $\alpha = 61$ cm & \multirow{3}{*}{$D_R = 0.862$} \\\cline{1-1}
\rule{0pt}{2.5ex} $\beta = 0$ cm & \\\cline{1-1}
\rule{0pt}{2.5ex} $\delta = 122$ cm & \\\hline
\multicolumn{2}{|c|}{\rule{0pt}{2.5ex} \textbf{Reduction in Drag}} \\\hline
\multicolumn{2}{|c|}{\rule{0pt}{2.5ex} $R_D = 13.8\%$} \\\hline
\end{tabular}
\begin{tabular}[t]{|c|c|}
\hline
\multicolumn{2}{|c|}{\rule{0pt}{2.5ex} \textbf{Class II Fairing} \\\hline
\rule{0pt}{2.5ex} \textbf{Configuration} & \textbf{Drag Ratio} \\\hline
\rule{0pt}{2.5ex} $W = 73^{\circ}$ & \multirow{4}{*}{$D_R = 0.826$} \\\cline{1-1}
\rule{0pt}{2.5ex} $X = 49^{\circ}$ & \\\cline{1-1}
\rule{0pt}{2.5ex} $Y = 19^{\circ}$ & \\\cline{1-1}
\rule{0pt}{2.5ex} $Z = 32^{\circ}$ & \\\hline
\multicolumn{2}{|c|}{\rule{0pt}{2.5ex} \textbf{Reduction in Drag}} \\\hline
\multicolumn{2}{|c|}{\rule{0pt}{2.5ex} $R_D = 17.4\%$} \\\hline
\end{tabular}
\end{table}
\end{document}
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Ok, facing that @StefanH found your missing }, and building on @Mico's solution, I tried expanding the topic, by refactoring your table. I added the following packages to your solution:
array-> to have\extrarowheightbooktabs-> for fancy rulessiunitxto typeset units with the proper spacing.
Furthermore, in order to avoid the abundance of $ I defined a new column type that automatically inserts that for you.
So, after a bit of clean up, here is the code, followed by the result. You will notice that I used fewer lines, since this way the information feel better grouped and more readable to me:
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{array,booktabs,multirow}
\usepackage{siunitx}
\title{Test}
\author{Chad Stucki}
\date{September 2016}
\newcolumntype{C}{>{$}c<{$}}
\begin{document}
\maketitle
\section{Introduction}
\begin{table}[!b]
\setlength\extrarowheight{2pt}
\centering
\caption{Caption}
\begin{tabular}[t]{Cc}
\toprule
\multicolumn{2}{c}{ \textbf{Class I Fairing}} \\
\multicolumn{1}{c}{\textbf{Configuration}} & \textbf{Drag Ratio} \\\midrule
\alpha = \SI{61}{\centi\metre}& \multirow{3}{*}{$D_R = 0.862$} \\
\beta = \SI{0}{\centi\metre} & \\
\delta = \SI{122}{\centi\metre}& \\\midrule
\multicolumn{2}{c}{ \textbf{Reduction in Drag}} \\
\multicolumn{2}{c}{ $R_D = 13.8\%$} \\\bottomrule
\end{tabular}
\begin{tabular}[t]{Cc}
\toprule
\multicolumn{2}{c}{ \textbf{Class II Fairing}} \\
\multicolumn{1}{c}{\textbf{Configuration}} & \textbf{Drag Ratio} \\\midrule
W = \ang{73} & \multirow{4}{*}{$D_R = 0.826$} \\
X = \ang{49} & \\
Y = \ang{19} & \\
Z = \ang{32}& \\\midrule
\multicolumn{2}{c}{ \textbf{Reduction in Drag}} \\
\multicolumn{2}{c}{ $R_D = 17.4\%$} \\\bottomrule
\end{tabular}
\end{table}
\end{document}
Moriambar
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Nice answer. I would add on empty row to the first table, or even better, merge both table in one :). But this doesn't matter now, OP is not respond yet (after nine months :-( ). – Zarko Jul 13 '17 at 19:39

}missing after\textbf{Class II Fairing}. And you need\usepackage{multirow}. – StefanH Sep 29 '16 at 18:58\rule{0pt}{2.5ex}directives and instead (b) load thearraypackage and issue the instruction (only once!)\setlength\extrarowheight{2pt}immediately after\begin{table}. – Mico Sep 29 '16 at 19:01