See, if the following result ids what you looking for:
\documentclass{article}
\usepackage{mathtools}
\usepackage{geometry}
\begin{document}
\begin{align}
d(0,0) & = e(0,0) \\
d(0,j) & = e(0,j) + d(0,j-1);\quad j = 1,2,\dots, N_\tau -1 \\
d(i,0) & = e(i,0) ;\quad i = 1,2,\dots, N_t -1 \\
%
d(1,j) & = e(1,j) + \begin{cases}
d(0,j) & \\
d(0,j-1) & j = 1,2,\dots, N_\tau -1 \\
d(1,j-1)) &
\end{cases} \\
d(i,1) & = e(i,1) + \begin{cases}
d(i,0) & \\
d(i-1,0) & i = 1,2,\dots, N_t -1 \\
d(i-1,1)) &
\end{cases} \\
d(i,j) & = e(i,j) + \begin{cases}
d(i-1,j-1) & \\
d(i-1,j-2) + e(i,j-1) & i > 1 \text{ and } j > 1 \\
d(i-2, j-1) + e(i-1,j) &
\end{cases}
\end{align}
\end{document}
Your MWE is unnecessary complicated, but in the same time there are missing ampersands and as say johannes-b in his comments, not allowed empty lines ...
Best Answer
You can use an
array
environment: