Realization of a stochastic process Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let's define a stochastic process as a function 
$$
S: \Omega \times \mathbb{R} \rightarrow \mathbb{R} \\
\omega \times t \mapsto S(\omega, t) \, .
$$
It makes sense that at fixed $t$, 
$$X_t(\cdot) \equiv S(\cdot, t)$$ 
is by definition a random variable. However I don't understand what is the meaning of 
$$X_\omega(\cdot) \equiv S(\omega, \cdot) \, .$$ 
An example could help. How do we fix $\omega$ in, for instance, a random walk or in the Gaussian noise? If we fix $\omega$, doesn't it imply that we will always get the same value for $X_\omega(\cdot)$?
 A: If we treat $\omega$ as a constant, then $S(\omega, \cdot): \mathbb{R} \to \mathbb{R}$ is a function. For example, any set of time series data such as a set of stock price data at hand is a "function of time", which is mathematically viewed as a realization of a stochastic process. To see this, recall what the horizontal axis measures and what the vertical axis measures in the stock price data set.
Note that the set $\Omega$ is simply an abstract space, whose elements need not be numbers. A typical example is the coin-tossing one. Tossing a fair coin can give us either the result "head" or the result "tail". So here we may take $\Omega := \{ \text{''head", "tail"} \}$. But can math speak something directly from $\Omega$? I am afraid not so. But with the help of the concept of random variable, which is a "nice" function on $\Omega$ in $\mathbb{R}^{n}$, math starts working. 
A phrase such as "we fix $\omega$" is a mathematical one, which does not mean that any one of us did manually somehow "determine" a value of $\omega$ in whatever sense you probably are thinking of :).
