Why doesn't \pagestyle{empty} work on the first page of a chapter?

The page style for the first page of a chapter is set internally (in the book and report document classes) to be plain; you can change this behaviour by adding the following lines to the preamble of your document:

\makeatletter
\renewcommand\chapter{\if@openright\cleardoublepage\else\clearpage\fi
                    \thispagestyle{empty}% original style: plain
                    \global\@topnum\z@
                    \@afterindentfalse
                    \secdef\@chapter\@schapter}
\makeatother

or, using the etoolbox package:

\usepackage{etoolbox}
\patchcmd{\chapter}{plain}{empty}{}{}

If the plain style is not needed elsewhere, then you can redefine it to be the empty page style; this can be done with:

\makeatletter
  \let\ps@plain\ps@empty
\makeatother

Finally, in the scrbook document class (from the KOMA-Script bundle), the style for the first page of chapters can be changed simply by redefining the \chapterpagestyle command, as in:

\renewcommand*\chapterpagestyle{empty}

EDIT: added egreg's remarks.

The \chapter command internally uses \thispagestyle{plain}. Add \thispagestyle{empty} immediately after \chapter.

\documentclass{book}

\pagestyle{empty}

\begin{document}

\chapter{The first}
\thispagestyle{empty}

This page has a page number\ldots

\newpage

\ldots but not this one.

\end{document}

The reason this happens is that the \chapter macro resets the pagestyle for that page. It makes sense when you think about it since you usually want the first page of a chapter to look different than other pages. For instance, if the page number normally sits on the top of the page, you will want to move it to the bottom of the chapter title page. It would look strange otherwise.

There are a number of fixes for the problem, but the easiest is to put \thispagestyle{empty} immediately after the \chapter macro.

If you'd rather change every chapter page, you could use the memoir class and redefine the chapter pagestyle:

\makepagestyle{chapter}

See also this question or this one.