Left align in math environment with every element over each other

Here is a possible solution if no equation number is required:

\documentclass{article}

\usepackage{mathtools}

\begin{document}

\begin{alignat*}{4}
  \gamma(0) &= \alpha_1\gamma(1)   &&+ \alpha_2\gamma(2)   &&+ \dots + \alpha_p\gamma(p)  &&+ \sigma^2 \\
  \gamma(1) &= \alpha_1\gamma(0)   &&+ \alpha_2\gamma(1)   &&+ \dots + \alpha_p\gamma(p-1)&&           \\
            &\vdotswithin{=} \\
  \gamma(p) &= \alpha_1\gamma(p-1) &&+ \alpha_2\gamma(p-2) &&+ \dots + \alpha_p\gamma(0)  &&
\end{alignat*}

\end{document}

If the centered equation number is indeed required, use an equation--alignedat combination, as in Thorsten's answer.

If you want to keep one single equation number, you can use the alignedat environment inside equation.

\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage{mathtools}  % loads »amsmath«

\pagestyle{empty}
\begin{document}
  \begin{equation}
    \begin{alignedat}{5}
      \gamma(0) &= \alpha_1\gamma(1) &&+ \alpha_2\gamma(2) &&+ \cdots &&+ \alpha_p\gamma(p)&&+ \sigma^2 \\
      \gamma(1) &= \alpha_1\gamma(0) &&+ \alpha_2\gamma(1) &&+ \cdots &&+ \alpha_p\gamma(p-1) \\
      &\vdotswithin{=} \\
      \gamma(p) &= \alpha_1\gamma(p-1)&&+ \alpha_2\gamma(p-2) &&+ \cdots &&+\alpha_p\gamma(0)
    \end{alignedat}
  \end{equation}
\end{document}

---

The accepted answer is definitely the way to go, but if ever you're stuck on a desert island and you don't have the amsmath package, an array environment can do some of the work- there are deficiencies (such as the alignment of the \vdots and the need for extra {}), but it does get some of the job done.

% arara: pdflatex
% !arara: indent: {overwrite: on}
\documentclass{article}
\begin{document}
\[
    \begin{array}{r@{}l@{}l@{}l@{}l@{}l@{}l@{}}
        \gamma(0) & {}=\alpha_1\gamma(1)    & {}+\alpha_2\gamma(2)   & {}+{}\cdots {} &   & {}+\alpha_p\gamma(p)   & {}+\sigma^2 \\
        \gamma(1) & {} =\alpha_1\gamma(0)   & {}+\alpha_2\gamma(1)   & {}+{}\cdots {} &   & {}+\alpha_p\gamma(p-1) &             \\
                  & \vdots                  &                        &                &   &                        &             \\
        \gamma(p) & {} =\alpha_1\gamma(p-1) & {}+\alpha_2\gamma(p-2) & {}+{}\cdots {} &   & {}+\alpha_p\gamma(0)   &
    \end{array}
\]
\end{document}