LaTeX enumerate - bold item with non-bold text

You can use the enumitem package. If you don't want to use additional packages, you can simply redefine \labelenumi:

\documentclass{article}
\usepackage{enumitem}

\begin{document}

\begin{enumerate}
    \renewcommand\labelenumi{\bfseries\theenumi}
    \item First item.
    \item Second item.
    \item Third item.
\end{enumerate}

% requires the enumitem package
\begin{enumerate}[label=\bfseries\arabic*]
    \item First item.
    \item Second item.
    \item Third item.
\end{enumerate}

\end{document}

EDIT: on a more programmatic level, you could define a new list-like environment that behaves like the standard enumerate, but with the desired format for the label. This can be done using something like this

\documentclass{article}

\newenvironment{boenumerate}
  {\begin{enumerate}\renewcommand\labelenumi{\textbf\theenumi}}
  {\end{enumerate}}

\begin{document}

\begin{boenumerate}
    \item First item.
    \item Second item.
    \item Third item.
\end{boenumerate}

\end{document}

or imitating the definition of the enumerate environment as given in source2e):

\documentclass{article}

\makeatletter
\def\boenumerate{%
\renewcommand\labelenumi{\textbf\theenumi}
  \ifnum \@enumdepth >\thr@@\@toodeep\else
    \advance\@enumdepth\@ne
    \edef\@enumctr{enum\romannumeral\the\@enumdepth}%
      \expandafter
      \list
        \csname label\@enumctr\endcsname
        {\usecounter\@enumctr\def\makelabel##1{\hss\llap{##1}}}%
  \fi}
\let\endboenumerate =\endlist
\makeatother

\begin{document}

\begin{boenumerate}
    \item First item.
    \item Second item.
    \item Third item.
\end{boenumerate}

\end{document}

Something similar, but now with the help of the enumitem package:

\documentclass{article}
\usepackage{enumitem}

\newlist{boenumerate}{enumerate}{4}
\setlist[boenumerate,1]{label=\bfseries\arabic*}

\begin{document}

\begin{boenumerate}
    \item First item.
    \item Second item.
    \item Third item.
\end{boenumerate}

\end{document}

You should use a package like enumitem. The package allows to modify the item by a key-value-syntax.

\documentclass{article}
\usepackage{enumitem}
\begin{document}
\begin{enumerate}[label=\textbf{\arabic*},start=0]
   \item First item
   \item Second item
   \item Third item
\end{enumerate}
\end{document}

It's not a good idea to redefine \item, as it's used in many parts of LaTeX. Here is a \benumerate environment in which you can use the \bolditem command alternating it to the normal one.

\documentclass[a4paper]{article}

\let\ifbolditem\iffalse
\def\bolditemtrue{\global\let\ifbolditem\iftrue}
\def\bolditemfalse{\global\let\ifbolditem\iffalse}

\newenvironment{benumerate}
  {\def\bolditem{\bolditemtrue\item}%
   \enumerate
   \renewcommand\makelabel[1]{\hss\llap{\ifbolditem\bfseries\fi##1\bolditemfalse}}}
  {\endenumerate}


\begin{document}
\begin{benumerate}
\item Normal
\bolditem Bold
\item Normal again
\bolditem[9] Bold
\end{benumerate}
\end{document}