How does the concept of a random variable come into play when sampling? I have an (intuitive) understanding of what a random variable and a sample are. However, I am not sure how a random variable is related to a sample. 
Suppose we have a population of people. Sampling from that population means to take a subset (which we call "sample") of those people. 
What can be considered a random variable while sampling, once we have a sample and when we take an element from this sample? How many random variables are involved in the process of sampling or once we have the sample?
I would appreciate seeing many real-world examples of the sampling process where you explain which and how many random variables are involved.

Notes

1. This question is different from What is the difference between random variable and random sample?.


2. Please, avoid defining what a random variable and sample are. I already know what they are and, if I forget the definitions, I can look them up again!

 A: There are two major ways of approaching random sampling. One is parametric, assuming that each unit responds based on a function of their characteristics plus an error. This error is the random variable that @Robien1 describes.
The other method treats the responses received from units in the population as fixed constants. The random variable in this case is a 0/1 indicator variable for each unit in the population, indicating whether the unit is selected in the sample or not.
In survey analysis the first method is called the model-based, the second is called the design-based or model-assisted. The design-based method is useful if you have access to (or can build) a population frame.
A: Let us take a  human population of $N$ persons. 
Among that population $N_0$ persons have blue eyes and let's call $p=\dfrac{N_0}{N}$ the proportion of blue-eyes persons.
If you pick « randomly » somebody in that population and associate the value 1 if that person has blue eyes and 0 otherwise, you are constructing a real random variable $X$ because il has uncertain issue.
To make it simple $X$ takes two values and $P(X=1)=p$, $P(X=0)=1-p$ ($P$ being  a measure of probability, the existence/construction of such  a probability is a big step, not necessary here, where the intuition makes it understable/acceptable).
For your sample, the $i$-th person picked is associated with a random variable $X_i$. All the $X_i$ follow the same law (of probability) called $B(p)$ (Bernoulli) : 1 with probability $p$, 0 with probability $1-p$).
So if you have a $n$-sample, you have a $n$ random variables associated
and what you handle in your real sample are issues of that variables: $X_1(\omega)=0$,
$X_2(\omega)=1$,
$\dots$,
$X_n(\omega)=0$ (for example)
Then the theory of probabilities give you some information
like $\dfrac{\sum\limits_{i=1}^nX_i}{n}\to p$ in a sense
if the variables are independant which in that case would be picking 
each time a new personn in the whole population, otherwise they are dependant
but if the $n$-sample  picked is small in comparison with $N$ then you can approximate with independance.
To finish you get :
$x_1,\dots x_n$ in your sample
and the probability model consider it as
$X_1(\omega),X_2(\omega),\dots,X_n(\omega)$
where the $X_i$ follow the same law of probability and are most of the time mutually independants (or so-approximated)
I hope it answers your question.
