In artcile.cls
without the titlepage
option, the definition of \maketitle
contains
\thispagestyle{plain}
You can discover this by issuing \show\maketitle
and looking the .log
file, but the output is quite long.
This means that either you need to change the plain
layout, or much simpler, you need to issue \thispagestyle{empty}
after \maketitle
has had effect, but before the page ends.
In your example, the latter is sufficient.

\documentclass[12pt]{article}
\pagestyle{empty} % no page number
\begin{document}
\title{Article name}
\author{Author}
\maketitle
\thispagestyle{empty}
\end{document}
If you had used the titlepage
option then there would have been no page number.
Note that other places where \thispagestyle{plain}
in article.cls
is issued is on the first page of an index if you have one.
See Page number on Index page for defining the plain
page style.
\thispagestyle{empty}
just after\maketitle
.