I'm surprised you didn't get an error like this:
pdflatex> ! Package natbib error: Bibliography not compatible with author-year citations.
the vancouver style is not compatible with natbib. Now, you have two possibilities:
1) Use natbib and try to find a bibliography style that somewhat resembles vancouver
2) Simply remove \usepackage{natbib} of your code and you'll be able to use \bibliographystyle{vancouver}.
Example by following 2):
\documentclass{article}
\usepackage[english]{babel}
\usepackage{url}
%\usepackage{natbib}
\begin{document}
Hello\cite{mario2013}
\bibliographystyle{vancouver}
\bibliography{refs}
\end{document}
refs.bib file looks like this:
@article{mario2013,
author={mario},
title={Marito and Friends},
journal={My Journal},
year = {2013},
}
And the output should look like this:

If you want to use author-year citations, you could load the natbib package along with a bibliography style of your choice (e.g. IEEEtranSN):
\documentclass{article}
\usepackage[english]{babel}
\usepackage{url}
\usepackage{natbib}
\begin{document}
According to \citet{mario2013} this is what the IEEEtranSN should look like
\bibliographystyle{IEEEtranSN}
\bibliography{refs}
\end{document}
The output will look something like this:
