Solid black box in the proof environment

The documentation of amsthm says:

A QED symbol, □, is automatically appended at the end of a proof environment. To substitute a different end-of-proof symbol, use \renewcommand to redefine the command \qedsymbol.

Thus, one might proceed as follows:

\renewcommand{\qedsymbol}{$\blacksquare$}

A mwe to play with:

\documentclass[a4paper,11pt]{article}
\usepackage{amsmath,amssymb,amsthm}

\renewcommand{\qedsymbol}{$\blacksquare$}

\begin{document}
\begin{proof}[Proof of the Main Theorem]
  \begin{equation*}
    G(t)=L\gamma!\,t^{-\gamma}+t^{-\delta}\eta(t) \qedhere
  \end{equation*}
\end{proof}
\end{document}

If you're looking for beamer, it is quite the same: you have to re-define the template qed symbol (almost similar to my previous answer in How to customise (e.g. make thicker) the end-proof symbol in Beamer?).

A mwe:

\documentclass{beamer}

\begin{document}

\begin{frame}
\setbeamertemplate{qed symbol}{$\blacksquare$}
\begin{proof}
\begin{itemize}
\item First item.
\item Second item.
\item Third item.
\end{itemize}
\end{proof}
\end{frame}

\end{document}

Notice that in this case the symbol will assume the colors of the Beamer color theme currently in use.

If you are using the amsthm package (or one of the ams documentclasses), then the symbol used is given by the command \qedsymbol. It can be redefined as follows:

\documentclass{article}

\usepackage{amsthm,amssymb}

\renewcommand{\qedsymbol}{\rule{0.7em}{0.7em}}

\begin{document}
\begin{proof}
  Clear.
\end{proof}
\end{document}