Inserting a picture in a title page

It's most easy with the titling package and its maketitlehookxcommands that allow for inserting supplementary material for the \maketitle command. Here is an example:

\documentclass[11pt,a4paper,twoside]{report}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[osf]{garamondx}
\usepackage[showframe, nomarginpar]{geometry}
\usepackage{graphicx}
\pagestyle{plain}

\usepackage{titling}
\title{About Piero di Cosimo}
\author{The author}

\setlength\droptitle {-38.5mm}
\pretitle{\begin{center}\Huge\itshape\bfseries}
\posttitle{\end{center}\vskip2cm}
\preauthor{\begin{center}\LARGE}
\postauthor{\end{center}\vfill}
\renewcommand{\maketitlehooka}{%
\noindent\includegraphics[width=\linewidth]{Piero_di_Cosimo_1}\par\vskip 3cm}%

\begin{document}

\maketitle

\end{document} 

!

There is no requirement that \includegraphics is in a figure environment, which is for including floating objects (including pictures) with their caption.

For your problem, use the titling package that allows customizing the \maketitle command adding items where needed.

\documentclass{article}
\usepackage{graphicx}
\usepackage{titling}

% set up \maketitle to accept a new item
\predate{\begin{center}\placetitlepicture\large}
\postdate{\par\end{center}}

% commands for including the picture
\newcommand{\titlepicture}[2][]{%
  \renewcommand\placetitlepicture{%
    \includegraphics[#1]{#2}\par\medskip
  }%
}
\newcommand{\placetitlepicture}{} % initialization

\begin{document}

\title{A very important paper}
\author{A. Uthor}
\date{38 July 2014}
\titlepicture[width=3in]{example-image}

\maketitle

\end{document}

So long as you have \titlepicture before \maketitle you'll get the picture where you want it. Not specifying \titlepicture will do nothing different from the usual \maketitle.

You can add it directly to the \author:

\documentclass{article}
\usepackage{graphicx}

\begin{document}

\title{A very important paper}
\author{A. Uthor\\[2ex]
        \includegraphics[width=3in]{example-image}\\[1ex]
        \small This is my title picture
        }
\date{38 July 2014}

\maketitle


\end{document}