Question: Given a probability space ($\Omega, \mathcal{F}, P$) and a measurable space ($S,\Sigma$), and an S-valued stochastic process { $X\_t : t \in T$ }(assume every $X\_t$ is i.i.d). What does a sample paths mean? From the definition, it's 'the function on $T$ to the range of the process which assigns to each $t$ the value $X\_t(\omega)$, where $\omega$ is a previously given fixed point in the domain of the process.

But if the $\omega$ is fixed, then the value of each $X\_t$ should be the same, which seems not true. What's the problem here?

Answer: 
The problem is that it's simply not true that "then the value of each $X\_j$ should be the same". 

Hmm. Example: Say $\Omega=[0,1]^2$ with $P$ equal to Lebesgue measure. Define $X,Y:\Omega\to\Bbb R$ by $X(s,t)=t$ and $Y(s,t)=s$. Then $X$ and $Y$ are iid, but $X(\omega)\ne Y(\omega)$.
Each random variable $X\_t$ is an $\mathcal F$-measurable function $\omega\mapsto X\_t(\omega)$ from $\Omega$ to $S$. The stochastic process $X$ may therefore be viewed as a function of two variables $\Omega\times T\ni (\omega,t)\mapsto X\_t(\omega)\in S$. For each fixed $\omega\in\Omega$ we have the partial function $t\mapsto X\_t(\omega)$; this is the *sample path* of $X$ corresponding to the sample point $\omega$.