Please, consult a beginner's manual for LaTeX. See What are good learning resources for a LaTeX beginner?
You mustn't use $ inside equation and that's the reason for the errors you get: equation already uses math mode.
Also, \emph is not for math formulas; single letters will be automatically typeset in italics.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}\label{NEW}
\sigma_{i} = \upsilon_{p}\rho(\dot{u}^{ff}-\dot{u}_{i}^{m})
\end{equation}
\end{document}

If ff is to be a single symbol, you might use \mathit{ff}
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}\label{NEW}
\sigma_{i} = \upsilon_{p}\rho(\dot{u}^{\mathit{ff}}-\dot{u}_{i}^{m})
\end{equation}
\end{document}

\emphin math mode? Math mode letters are italicized by default. Does\sigma_i = \upsilon_p \rho (\dot{u}^{ff} - \dot{u}_{i}^{m})do what you want? If not, post a minimal working example. – frabjous Jun 25 '22 at 14:49equationestablishes math mode, and using$within display math is incorrect -- it turns off math mode and changes many processing conventions (also causing many additional irrelevant error messages). Also,\emphshouldn't be used within math mode; letters within math are by default italic. – barbara beeton Jun 25 '22 at 14:52