Adding a caption to an equation in LaTeX

You may want to look at http://tug.ctan.org/tex-archive/macros/latex/contrib/float/ which allows you to define new floats using \newfloat

I say this because captions are usually applied to floats.

Straight ahead equations (those written with $ ... $, $$ ... $$, begin{equation}...) are in-line objects that do not support \caption.

This can be done using the following snippet just before \begin{document}

\usepackage{float}
\usepackage{aliascnt}
\newaliascnt{eqfloat}{equation}
\newfloat{eqfloat}{h}{eqflts}
\floatname{eqfloat}{Equation}

\newcommand*{\ORGeqfloat}{}
\let\ORGeqfloat\eqfloat
\def\eqfloat{%
  \let\ORIGINALcaption\caption
  \def\caption{%
    \addtocounter{equation}{-1}%
    \ORIGINALcaption
  }%
  \ORGeqfloat
}

and when adding an equation use something like

\begin{eqfloat}
\begin{equation}
f( x ) = ax + b
\label{eq:linear}
\end{equation}
\caption{Caption goes here}
\end{eqfloat}

As in this forum post by Gonzalo Medina, a third way may be:

\documentclass{article}
\usepackage{caption}

\DeclareCaptionType{equ}[][]
%\captionsetup[equ]{labelformat=empty}

\begin{document}

Some text

\begin{equ}[!ht]
  \begin{equation}
    a=b+c
  \end{equation}
\caption{Caption of the equation}
\end{equ}

Some other text
 
\end{document}

More details of the commands used from package caption: here.

A screenshot of the output of the above code:

The \caption command is restricted to floats: you will need to place the equation in a figure or table environment (or a new kind of floating environment). For example:

\begin{figure}
\[ E = m c^2 \]
\caption{A famous equation}
\end{figure}

The point of floats is that you let LaTeX determine their placement. If you want to equation to appear in a fixed position, don't use a float. The \captionof command of the caption package can be used to place a caption outside of a floating environment. It is used like this:

\[ E = m c^2 \]
\captionof{figure}{A famous equation}

This will also produce an entry for the \listoffigures, if your document has one.

To align parts of an equation, take a look at the eqnarray environment, or some of the environments of the amsmath package: align, gather, multiline,...