Intuition in Random walk Suppose $X_i$ are i.i.d. r.v.
$S_n=X_1+\cdots+X_n$ is random walk.
Why $\mathcal{F}_n =\sigma(X_1,\cdots,X_n)$ are called the information known at time n?
I think We only know the measurability of such form:{$X_i<m$} for $i\leq n$.What's the meaning of information?
Further more,if N is a stopping time.
$\mathcal{F}_N=\{A|A\cap\{N=n\}\in\mathcal{F}_n,for\quad n<\infty$}= the information known at time N.
How to understand this?
More questions:(in the definition of $\mathcal{F}_N$)
(1)What is the whole space where $A $ from?$\mathcal{F}_\infty?$
(2)Why "$for\quad n<\infty$" not $\leq$?
 A: You can see $\sigma$-algebras as gathering of information. Following your example, knowing $\mathcal{F}_{n}$ is entirely sufficient to know the value of $X_{1}, ..., X_{n}$ for every $\omega$ : if you give me some $\omega $ and if I know $\mathcal{F}_{n}$, I'll be able to find a (measurable) subset of the form $\{X_{n} = x\}$ and therefore I'll be sure that $X_{n}(\omega)=x$ without even knowing some expression to compute $X_{n}$. 
On the other side, if I only know $\mathcal{F}_{n}$, I won't be able to find the value of $X_{n+1}$ since I have no access to the sets of the form $\{ X_{n+1} \in A \}$, with $A$ borel set.
You can find an excellent explanation of this is Probability and stochastics by Ehran Cinlar. 
Let follow this intuition for stopping times. You have to see $T$ as an alarm. Let $A$ be an event (which means it lives in $\mathcal{F}_{\infty}$, the backgound $\sigma$-algebra of the processus). It need not be in $\mathcal{F}_{n}$ for a particular $n$ : for example, it can be an event of the form $\{\lim X_{n} = \infty \}$, which is in $\mathcal{F}_{\infty}$.
The set $A \cap \{T=n\}$ means "$A$ is happening and the alarm is ringing at time $n$". Suppose you're at a time $n$ and you know the alarm just rang. You're therefore on the set $\{T=n\}$. The statement $A\cap\{T=n\}\in \mathcal{F}_{n}$ means that you have access to the event $A$, even if $A$ might not be in $\mathcal{F}_{n}$, precisely because the alarm rings. You are precisely on the region of $A$ that can be known when the alarm rings. If the alarm hasn't rung yet, it's impossible for you to check if $A$ is happening.
That's why we call $\mathcal{T}_{t}$ the "$\sigma$-algebra from the past".
These kind of things take time to master. Studying processes and martingales will certainly get you acquainted with it.
