Why does the following generate an error?
\documentclass{article}
\usepackage{enumitem}
\begin{document}
\begin{description}[label=\emph{\alph*})]
\item [aaa] bbbb
\item [eee] əəəə
\end{description}
\end{document}
Why does the following generate an error?
\documentclass{article}
\usepackage{enumitem}
\begin{document}
\begin{description}[label=\emph{\alph*})]
\item [aaa] bbbb
\item [eee] əəəə
\end{description}
\end{document}
Description lists don't admit a label in the sense of the label key of the enumitem package.
According to a comment to the original question, the purpose is to get description items numbered; in this case, you can use an enumerate environment (setting the label key appropriately to get the desired numbering) and a variant of \item to emulate the format of a description list; something along these lines
\documentclass{article}
\usepackage{enumitem}
\newcommand\litem[1]{\item{\bfseries#1.\space}}
\begin{document}
\begin{enumerate}[label=\emph{\alph*)}]
\litem{Term one} First term description.
\litem{Term two} Second term description.
\end{enumerate}
\end{document}

The \item of description lists must be formatted with enumitem's font key (or its format synonym). Its last command can take an argument, so here's how to add a closing parenthesis after the item name:
\documentclass{article}
\usepackage{enumitem}
\newcommand*{\textitplusparen}[1]{\textit{#1)}}
\begin{document}
\begin{description}[font=\normalfont\textitplusparen]
\item [Item name] Some text
\end{description}
\end{document}

You can even use \item, provided you do some juggling of the macros:
\documentclass{article}
\usepackage{enumitem}
\newcommand{\changeitem}{%
\let\latexitem\item
\renewcommand\item[1][]{\latexitem\relax{\bfseries##1} }%
}
\newenvironment{descenum}[1][]
{\begin{enumerate}[before=\changeitem,#1]}
{\end{enumerate}}
\begin{document}
\begin{descenum}
\item[Term one] First term description.
\item[Term two] Second term description.
\end{descenum}
\end{document}
The descenum environment accepts all kind of options good for enumerate (as provided by enumitem), for instance
\begin{descenum}[label=(\arabic*)]
\item[Term one] First term description.
\item[Term two] Second term description.
\end{descenum}