The equations do not align with respect to +

You need one more & to have stuff left-aligned after it (if you want to work with aligned here).

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[\begin{aligned}
p       &= q r             &&+ r_0 \\
r       &= q_0 r_0         &&+ r_1 \\
r_0     &= q_1 r_1         &&+ r_2 \\
r_k     &= q_{k+1} r_{k+1} &&+ r_{k+2} \\
        &\mathrel{\vdots} \\
r_{k+1} &= q_{k+2} r_{k+2} \\
\end{aligned}\]
\end{document}

If you want to center the \vdots, you can use the mathtools package.

\documentclass{article}
\usepackage{mathtools}
\begin{document}
\[\begin{aligned}
p       &= q r             &&+ r_0 \\
r       &= q_0 r_0         &&+ r_1 \\
r_0     &= q_1 r_1         &&+ r_2 \\
r_k     &= q_{k+1} r_{k+1} &&+ r_{k+2} \\
        &\vdotswithin{=} \\
r_{k+1} &= q_{k+2} r_{k+2} \\
\end{aligned}\]
\end{document}

You need alignedat, that doesn't add horizontal spacing, but with one & more.

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\[
\begin{alignedat}{2}
p       &= q r             &&+ r_0 \\
r       &= q_0 r_0         &&+ r_1 \\
r_0     &= q_1 r_1         &&+ r_2 \\
        &\mathrel{\;\vdots} \\
r_k     &= q_{k+1} r_{k+1} &&+ r_{k+2} \\
r_{k+1} &= q_{k+2} r_{k+2}
\end{alignedat}
\]

\end{document}