According to the book that I am reading,
A nonnegative, integer valued random variable T is a stopping time for the sequence {$Z_{n},n\geqslant0$} if the event T = n depends only on the value of random variables $Z_0 , Z_1,...,Z_n$. (It is known that the sequence $Z_1,Z_2,....,Z_n$ is a martingale).
Doubt 1: What does T = n mean?
I understand why the T can't be dependent on $Z_{n+1},Z_{n+2},...$ because stopping time must depend on the past. We cannot look into the future and decide that that particular point must be the stopping time.
Doubt 2: What I don't understand, and is written in the text is that we need to charecterize conditions on the stopping time T that maintain the property $E[Z_T] = E[Z_0]$. Why do we need this property?
To answer your first question, it is common probabilist short-hand to write things like $\{T = n\}$ to mean $\{\omega \in \Omega : T(\omega) = n\}$. Thus the event $\{T=n\}$ is the outcomes for which the stopping time evaluates to $n$.
For your second question, recall that a martingale satisfies $E[Z_n] = E[Z_0]$ for all $n$. It turns out that under certain circumstances you can replace the deterministic $n$ with a stopping time, and say $E[Z_T] = E[Z_0]$. This is great because it allows us to play in all new kinds of ways with martingales. The problem is that $E[Z_T] = E[Z_0]$ doesn't always hold, only under certain conditions (e.g. if there is a constant $N$ such that $T \leq N$ almost surely). So we would like to characterize when we can use this extension of the martingale property.