Let's say we want to model the fivefold throw of a fair coin. Then we define the corresponding process via the outcome of the throws, i.e. we set
$$X_t := \begin{cases} 1 & \text{t-th throw is head} \\ 0 & \text{otherwise} \end{cases}$$
for $t \in \{1,\ldots,5\}$. Now, since we have a fair coin, the probability that $X_t$ equals $1$ is 0.5 for each $t$. In probability theory, this is translated in the following abstract way: For a probability space $(\Omega,\mathcal{A},\mathbb{P})$, the random variables $X_t$ have to satisfy $\mathbb{P}(X_t=1)=\tfrac{1}{2}$. This means in particular that we do not care how the probability space looks like; only the distribution of the random variables is of importance.
Moreover, for any $\omega \in \Omega$ the mapping $t \mapsto X_t(\omega)$ is a realization of our process. If we throw the coin five times and observe e.g. $$0 \, \, 1 \, \, 0 \, \, 0 \, \, 1,$$ then there exists $\omega \in \Omega$ which "symbolizes" this outcome, i.e.
$$(X_1(\omega),X_2(\omega),X_3(\omega),X_4(\omega),X_5(\omega))=(0,1,0,0,1).$$
Usually, we are interested in questions like "What is the probability that we throw head 3 out of 5 times?"; this probability equals $\mathbb{P}(\sum_{t=1}^5 X_t = 3)$. This question can be answered if we know the (finite dimensional) distributions of the stochastic process $(X_t)_t$.
So, basically a stochastic process (on a given probability space) is an abstract way to model actions or events we observe in the real world; for each $\omega \in \Omega$ the mapping $t \mapsto X_t(\omega)$ is a realization we might observe. The likeliness of the realization is characterized by the (finite dimensional) distributions of the process.
It can help to look at some sample paths. From Bernt Øksendal's Stochastic Differential Equations:
The image shows five sample paths of a geometric Brownian motion process $\{X_t\}_{t\ge0}$. The paths are different functions of $t$ (which you may think of as representing time). Each one of them shows the values that $X$ takes over time under a specific outcome ($\omega_1,\dots,\omega_5$) in the sample space $\Omega.$
Notice by contrast that the expected value $E[X_t]$ of the process is also drawn (as a smooth function of $t$) and that it is not a sample path; rather than being the result of any single outcome, it is obtained by "averaging" over all of them according to the law of the process.
Best Answer
The step from finite case to infinite case is not so obvious and requires the very famous Kolmogorov extension theorem.