It is common in probability to define a prob. space as :
$$(\Omega,\mathscr{F},P)$$
This can be understood as sample space, events, and probabilities for each event. However I don't know how to to read/interpret when a probability space is defined as:
$$(\Omega,\mathscr{F},(\mathscr{F}_n)_{0\leq n\leq N},P)$$
What is $(\mathscr{F}_n)$ and how should this be interpreted?
For instance look at this definition for Stopping time:
On a discrete-time model build on a probability space $(\Omega,\mathcal{F},(\mathcal{F}_n)_{0\leq n\leq N},P)$ a random variable $v$ taking values in $\{0,1,2 ... N\}$ is a stopping time if, for any $n \in \{0,1,2 ... N\}$, $$\{v=n\}\in \mathscr{F}_n$$
Have my question above in mind what does this definition actually say about stopping time?
You can interpret $(\mathscr{F}_n)$ as the "knowledge" we have at "time" n. (Formally, this is a collection of $\sigma$-algebras, which give us our measurable sets, which defines the scope of measurable functions that we can have).
Now, we say a random variable is a stopping time if we can decide whether it is time to "stop" at time $n$ for any $n$. In the formal definition, it is saying to check whether $\{ \tau = n \}$ (the event to stop at time n) is in $\mathscr{F}_n$ (whether it is something we can indeed measure by time $n$).
(I don't know why your definition has a bound $N$ on it; that is not necessary to the best of my knowledge)