What exactly are random variables in probability theory? What I understood about random variables is :
We need to define a function that maps the set of possible outcomes of a random experiment to the 1-D real space. The notion of random variable is to represent an element in range space of the mapping.
What is this function called ?
Please correct me if I am understanding it wrong.
How exactly is the function defined if we have n trials. Can you please explain me how the function is different from 1 trial?
@zoli : The concept of two random variables even confused me a lot. I am writing my understanding of trial models please map it to the explanation you provided.  Suppose we have rolling a die example. Let A be an event of rolling the die. Outcomes of event A = { 1, 2, 3, 4, 5, 6}. We define event B as - to repeat A for 2 times (2 trials) and ask question like probability of getting two 6's. For this 36 possible outcomes. Let function1 for A would simply map occurrence of 6 to 1 and 0 otherwise. We will have the same function (function1) for the second trial also. Now, we have the sample space for B as { (1,0), (1,1), (0,1), (0,0) }. The function2 for event B could be map (i, j) to 1 if i, j both are 1, 0 otherwise. We will be interested in the value of outcomes to be 1 for function2. According to your explanation, I will have a tuple of two R's as a single element of range space of B. Please correct me. I am not getting the "vector of functions" thing.
 A: In probability theory we model the possible outcomes of an experiment (trial) with an abstract set whose elements are called elementary events. Certain sets of elementary events are the events. Without further specifying these sets, let's call them measurable sets. There is also a mapping from the measurable sets to $[0,1]$. This function is called probability.
The random variable is a measurable function. Measurability means that the inverse images of sets in the form $(-\infty,x)$ are measurable sets, that is the probability of them can be defined.
EXAMPLE
Consider the following elementary events:
$$\Omega=\{a,b,c\}$$
and the following random variable
$$f(\omega)=\begin{cases}1, \text{ if } \omega\in\{a,b\}\\
2, \text {otherwise}.
\end{cases}$$
So far we've been talking about one experiment (one trial).
Cartesian products of "one experiment models" can be used to model series of experiments. In such a case the random variables will form vectors of functions described in the firs paragraph.
Don't forget that even an infinitely long series of experiments can be considered as one trial, and one can ask how to model a new experiment consisting of, say, $3$ such trials (infinitely long series) of the original experiments. The method would be the same as described above.
Let' generate a "two trial model": Now,
$$\Omega'=\Omega\times\Omega=\{\omega':(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)\}.$$
We have two random variables $f_1$ depending only on the first argument of $\omega'$ and $f_2$ depending only on the second element of $\omega'$.
Now, try you have a vector 
$$f'(\omega')=\begin{bmatrix}f_1(\omega')\\
f_2(\omega')\end{bmatrix}.$$
