Standard document classes (like article and the like) typeset footnotes as a marker and text. The marker uses the footnote counter in terms of it's presentation. As such redefining this will redefine the way it displays. However, if you are only interested in modifying the marker display in the footnote itself and not in the actual text, you need to modify the footnote display function \@makefntext.
Using
\makeatletter\show\@makefntext\makeatother
produces
> \@makefntext=\long macro:
#1->\parindent 1em\noindent \hb@xt@ 1.8em{\hss \@makefnmark }#1
where #1 refers to the text of the footnote, and \@makefnmark refers to the typesetting of the actual footnote marker. Consequently, we can redefine this macro to suit our needs
\makeatletter%
\long\def\@makefntext#1{%
\parindent 1em\noindent \hb@xt@ 1.8em{\hss \textsuperscript(\kern-0.1ex\@makefnmark\kern-0.1ex\textsuperscript)}#1}
\makeatother
which will produce regular footnote markers in the text (without brackets ( )) but bracketed footnote markers in the footnotes ( ). Here's an example of the differences/similarities:

\documentclass{article}
\usepackage[paperheight=2in]{geometry}% http://ctan.org/pkg/geometry
\makeatother
\begin{document}
This is a piece of text\footnote{This is the first footnote}. \par
\begingroup
\renewcommand{\thefootnote}{(\arabic{footnote})}% Modify footnote globally
This is a piece of text\footnote{This is the first footnote}. \par
\endgroup
\begingroup
\makeatletter%
\long\def\@makefntext#1{%
\parindent 1em\noindent \hb@xt@ 1.8em{\hss \textsuperscript(\kern-0.1ex\@makefnmark\kern-0.1ex\textsuperscript)}#1}
This is a piece of text\footnote{This is the first footnote}.
\makeatother
\end{document}
Of course, using this approach, you can choose to use [ ] or { } as well if you'd like. geometry was only used to modify the page height for visual presentation.
()are "parentheses" not braces. Braces would be{}or maybe[]... – Seamus Jul 20 '11 at 15:10I wouldn't've used {[tag:formatting]} here, but you did -- may I ask why? I'd use {[tag:punctuation]}, and I'm thinking about widening the scope of the question to include the variants you provided in your answer, in order to make it canonical. What do you think? – doncherry Mar 28 '12 at 15:45