How to align the equal signs in this equation such that the first line and last 2 lines align

Here goes an answer:

\documentclass{amsart}
\begin{document}
\begin{align*} 
&CoJPoD_{   \{I_1, \dots, I_{k-1}, I_{k+1}, \dots, I_{n}\} \mid I_{k} }  \\ 
&= CoJPoD_{ \{\mathrm{system} \backslash I_{k}\} \mid I_k}  \\ 
&= P\bigl( X_1 \ge X_d^{I_1}, \dots, X_{k-1} \ge X_d^{I_{k-1}}, X_{k+1} \ge X_d^{I_{k+1}}, \dots, X_n \ge X_d^{I_n} \mid X_k \ge X_d^{I_k}\bigr) \\ 
&=\frac{P\bigl( X_1 \ge X_d^{I_1}, \dots, X_{k-1} \ge X_d^{I_{k-1}}, X_k \ge X_d^{I_k}, X_{k+1} \ge X_d^{I_{k+1}}, \dots, X_n \ge X_d^{I_n}\bigr)}{P\bigl(X_k \ge X_d^{I_k}\bigr)} \\ 
&= \frac{JPoD_{\{I_1, I_2, \dots, I_n\}}}{PoD_{I_k}}\\ 
&= \frac{JPoD_{\mathrm{system}}}{PoD_{I_k}} 
\end{align*}
\end{document}

Some of the changes I have made are:

• indicating how the equations should be aligned by using the & syntax.
• using \mathrm{text} for subscripts which are text of length more than 1.
• replace the \cdots directive by \dots so that amsclasses figure what is best to do.
• replace | by \mid.
• replace \left(...\right) by \bigl(...\bigr).

You'd find this entire document a very useful read. In particular, I'd like to draw you attention to the examples and discussion of the various align environments in Section 3 (pp. 3-9).

Output

I would still avoid the expression if possible because seemingly it's blurring the fact that the inequalities are repeated for the whole index set except k which should be easy to convey for the author and to grasp for the reader. Something along the lines of

\documentclass{article}
\usepackage{mathtools} %<- Loads amsmath and enhances/fixes it.

\begin{document}

Let us introduce the shorthand notation $\sigma_i$ for the 
inequality $X_i \ge X_d^{I_i}$. Then we have,
\begin{align*}
 CoJPoD_{   \{I_1, \cdots, I_{k-1}, I_{k+1}, \cdots, I_{n}\} | I_{k} }  &= CoJPoD_{ \{\text{system}\backslash I_{k}\} | I_k}\\
 &= P\left( \sigma_1, \cdots, \sigma_{k-1}, \sigma_{k+1}, \cdots, \sigma_{n} | \sigma_k\right) \\ 
 &= \frac{P\left( \sigma_{1}, \cdots, \sigma_n\right)}{P\left(\sigma_k\right)} \\
 &= \frac{JPoD_{\{I_1, I_2, \cdots, I_n\}}}{PoD_{I_k}} \\
 &= \frac{JPoD_{system}}{PoD_{I_k}}
\end{align*}
\end{document}

Last minute edit: I've missed the fix for the second system word and the ...PoDs. But other answers covered those pretty thoroughly.

And this is the solution, giving the output that you described.

\begin{align*}
 CoJPoD_{   \{I_1, \cdots, I_{k-1}, I_{k+1}, \cdots, I_{n}\} | I_{k} }  &= CoJPoD_{ \{system \backslash I_{k}\} | I_k}  \\
\noalign{\text{$ = P\left( X_1 \ge X_d^{I_1}, \cdots, X_{k-1} \ge X_d^{I_{k-1}}, X_{k+1} \ge X_d^{I_{k+1}}, \cdots, X_n \ge X_d^{I_n} | X_k \ge X_d^{I_k}\right) $}}\\
\noalign{\text{$ = \displaystyle\frac{    P\left( X_1 \ge X_d^{I_1}, \cdots, X_{k-1} \ge X_d^{I_{k-1}}, X_k \ge X_d^{I_k}, X_{k+1} \ge X_d^{I_{k+1}}, \cdots, X_n \ge X_d^{I_n}\right)         }{P\left(X_k \ge X_d^{I_k}\right)}$}} \\
& = \frac{JPoD_{\{I_1, I_2, \cdots, I_n\}}}{PoD_{I_k}} \\
& = \frac{JPoD_{system}}{PoD_{I_k}}
\end{align*}

However, I'd rather like the following code:

\begin{align*}
 \mathit{CoJPoD}_{   \{I_1, \ldots, I_{k-1}, I_{k+1}, \ldots, I_{n}\} | I_{k} }  &= \mathrm{CoJPoD}_{ \{\mathrm{system} \setminus I_{k}\} | I_k}  \\ %either \mathit or \mathrm
\noalign{\text{$ = P\left( X_1 \ge X_d^{I_1}, \dots, X_{k-1} \ge X_d^{I_{k-1}}, X_{k+1} \ge X_d^{I_{k+1}}, \ldots, X_n \ge X_d^{I_n} | X_k \ge X_d^{I_k}\right) $}}\\
\noalign{\text{$ = \displaystyle\frac{    P\left( X_1 \ge X_d^{I_1}, \ldots, X_{k-1} \ge X_d^{I_{k-1}}, X_k \ge X_d^{I_k}, X_{k+1} \ge X_d^{I_{k+1}}, \ldots, X_n \ge X_d^{I_n}\right)         }{P\left(X_k \ge X_d^{I_k}\right)}$}} \\
& = \frac{\mathrm{JPoD}_{\{I_1, I_2, \cdots, I_n\}}}{\mathrm{PoD}_{I_k}} \\ %either \mathit or \mathrm
& = \frac{\mathit{JPoD}_{\mathrm{system}}}{\mathrm{PoD}_{I_k}}  %either \mathit or \mathrm
\end{align*}

Certainly, CoJPoD, JPoD, PoD should be either \mathrm or \mathit. This gives us better spacing. See also usage of \setminus instead of \backslash and \ldots instead of \cdots.