Caption without text below "Figure''

You can use the caption package with the option format=hang to do this

\documentclass{article}
\usepackage[format=hang]{caption}

\begin{document}

\begin{figure}
    \centering
    \rule{20pt}{10pt} %just for demonstration
    \caption{ Here's a long caption, it goes across more than
    one line| the style you describe is called `hanging'.}
\end{figure}

\end{document}

Putting an enumerate environment in a caption can be done- see

Using the itemize environment inside a caption

for details.

\documentclass{article}
\usepackage[format=hang,labelsep=colon]{caption}    
\begin{document}

\begin{figure}
    \centering
    \rule{5cm}{3cm} %just for demonstration
    \caption[text for the list of figures]
            { \tabular[t]{@{}p{5cm}@{}}
              bla bla bla bla and some other nonsense text \\
              a) bla bla bla    \\
              b) bla bla bla
              \endtabular}
\end{figure}

\end{document}

Hanging indent is the default setup for captions in the classes from the KOMA-Script bundle.

\documentclass[paper=letter]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage{lipsum}

\begin{document}
  \begin{figure}[!ht]
    \centering
    \rule{3.2in}{1.8in}
    \caption[LoF entry]{\lipsum[1]}
  \end{figure}
\end{document}