How do I use tab stops in enumerate or itemize?

You can use the package tabto

\documentclass{article}
\usepackage{tabto}

\begin{document}
  \TabPositions{4cm}
  \begin{enumerate}[label=LEVEL \arabic{*},leftmargin=*]
    \item Description of Level 1 \tab L1 \\ \emph{more text}
    \item Description of Level 2 \tab L2 \\ \emph{more text} 
  \end{enumerate}
\end{document}

One can also use more tabs

\documentclass{article}
\usepackage{tabto}

\begin{document}

\NumTabs{3}

\begin{itemize}
\item some text           \tab i want this              \tab same here
\item another text        \tab and this to be at the    \tab same here
\item not the same length \tab same level after let say \tab seconde level here
\end{itemize}

\end{document}