problem with align

You missed a & in the first equation and a \\ before \shortintertext{short remark}

\documentclass{article}
\usepackage{mathtools}

\begin{document}
\begin{align} 
f_{**p}(X_p,Y_p)  &= X_p(Y(f)) = \sum_{i,j = 1} ^n \alpha_i
\beta_j E_{pi}(E_j(f)) f\\
\shortintertext{short remark}
E_{pi}(E_j(f)) &= \frac{\partial}{\partial x_i}(E_j(f) \circ \phi^{-1} )|_{\phi(p)} \\ 
               &= \frac{\partial}{\partial x_i} 
\end{align}
\end{document}

On request from Op:

You can use \span\omit as in:

\documentclass{article}
\usepackage{mathtools}

\begin{document}
\begin{equation}
f_{**p}(X_p,Y_p)  = X_p(Y(f)) = \sum_{i,j = 1} ^n \alpha_i
\beta_j E_{pi}(E_j(f)) f
\end{equation}
\begin{align} 
 f_{**p}(X_p,Y_p)  = X_p(Y(f)) = \sum_{i,j = 1} ^n \alpha_i
\beta_j E_{pi}(E_j(f)) f \span\omit\\
\shortintertext{short remark}
E_{pi}(E_j(f)) &= \frac{\partial}{\partial x_i}(E_j(f) \circ \phi^{-1} )|_{\phi(p)} \\ 
               &= \frac{\partial}{\partial x_i} 
\end{align}
\end{document}

If you don't want the first equation to align with the others, use:

\documentclass{article}
\usepackage{mathtools}

\begin{document}
\begin{gather}
  f_{**p}(X_p,Y_p)  = X_p(Y(f)) = \sum_{i,j = 1} ^n \alpha_i
\beta_j E_{pi}(E_j(f)) f
\\
\shortintertext{short remark}
\begin{align}
  E_{pi}(E_j(f)) &= \frac{\partial}{\partial x_i}(E_j(f) \circ \phi^{-1} )|_{\phi(p)} \\
               &= \frac{\partial}{\partial x_i}
\end{align}
\end{gather}
\end{document}