You can use tikZ for drawing curly brackets. For remembering the locations I defined the command \tikzmark, which is used inside the table cells.
In general, it is better to write table captions above the table (see Why should a table caption be placed above the table?).
To have the table caption below without overlapping with the brackets you can add \vspace{15mm} manually.
There may be a better (automatic) way.
Code:
\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{calc}
\newcommand{\tikzmark}[1]{\tikz[remember picture,overlay]\node[yshift=2pt](#1){};}
\begin{document}
\begin{table}
\begin{tabular}{l|l|l|l|l|l}
Weeks & $t - \Delta t $ & $\dotsm$ & $t-2$ & $t-1$ & $t$ \\
\hline
Search Volume & \tikzmark{a1}$n(t - \Delta t $ & $\dotsm$ & \tikzmark{b1}$n(t-2)$ & $n(t-1)$\tikzmark{a2} & $n(t)$\tikzmark{b2} \\
\end{tabular}
\begin{tikzpicture}[overlay, remember picture]
\draw [decorate,decoration={brace,amplitude=10pt,mirror,raise=4pt}] (a1.west) --node[below=14pt]{average = $N_{t-1, \Delta t}$} (a2.east);
\draw [decorate,decoration={brace,amplitude=10pt,mirror,raise=4pt}] ($(b1.west) - (0,1)$) --node[below=14pt]{$\Delta n_{t, \Delta t} = n_t - N_{t-1}, \Delta_t$} ($(b2.east) - (0,1)$);
\end{tikzpicture}
\vspace{15mm}
\caption{my caption}
\end{table}
\end{document}
Result:

\documentclass{...}, the required\usepackage's,\begin{document}, and\end{document}. That may seem tedious to you, but think of the extra work it represents for TeX.SX users willing to give you a hand. Help them help you: remove that one hurdle between you and a solution to your problem. – dexteritas May 28 '18 at 12:58