What is meant by a "random variable"? What do they mean when they say "random variable"? 
 A: Introduction
In thinking over a recent comment, I notice that all replies so far suffer from the use of undefined terms like "variable" and vague terms like "unknown," or appeal to technical mathematical concepts like "function" and "probability space."  What should we say to the non-mathematical person who would like a plain, intuitive, yet accurate definition of "random variable"?  After some preliminaries describing a simple model of random phenomena, I provide such a definition that is short enough to fit on one line.  Because it might not fully satisfy the cognoscenti, an afterward explains how to extend this to the usual technical definition.
Tickets in a box
One way to approach the idea behind a random variable is to appeal to the tickets-in-a-box model of randomness.  This model replaces an experiment or observation by a box full of tickets. On each ticket is written a possible outcome of the experiment.  (An outcome can be as simple as "heads" or "tails" but in practice it is a more complex thing, such as a history of stock prices, a complete record of a long experiment, or the sequence of all words in a document.)  All possible outcomes appear at least once among the tickets; some outcomes may appear on many tickets.
Instead of actually conducting the experiment, we imagine thoroughly--but blindly--mixing all the tickets and selecting just one.  If we can show that the real experiment should behave as if it were conducted in this way, then we have reduced a potentially complicated (and expensive, and lengthy) real-world experiment to a simple, intuitive, thought experiment (or "statistical model").  The clarity and simplicity afforded by this model makes it possible to analyze the experiment.
An example
Standard examples concern outcomes of tossing coins and dice and drawing playing cards.  These are somewhat distracting for their triviality, so to illustrate, suppose we are concerned about the outcome of the US presidential election in 2016.  As a (tiny) simplification, I will assume that one of the two major parties--Republican (R) or Democratic (D)--will win.  Because (with the information presently available) the outcome is uncertain, we imagine putting tickets into a box: some with "R" written on them and others with "D".  Our model of the outcome is to draw exactly one ticket from this box.
There is something missing: we haven't yet stipulated how many tickets there will be for each outcome. In fact, finding this out is the principal problem of statistics: based on observations (and theory), what can be said about the relative proportions of each outcome in the box?
(I hope it's clear that the proportions of each kind of ticket in the box determine its properties, rather than the actual numbers of each ticket.  The proportions are defined--as usual--to be the count of each kind of ticket divided by the total number of tickets.  For instance, a box with one "D" ticket and one "R" ticket behaves exactly like a box with a million "D" tickets and a million "R" tickets, because in either case each type is 50% of all the tickets and therefore each has a 50% chance of being drawn when the tickets are thoroughly mixed.)
Making the model quantitative
But let's not pursue this question here, because we are near our goal of defining a random variable.  The problem with the model so far is that it is not quantifiable, whereas we would like to be able to answer quantitative questions with it.  And I don't mean trivial ones, either, but real, practical questions such as "if my company has a billion Euros invested in US offshore fossil fuel development, how much will the value of this investment change as a result of the 2016 election?"  In this case the model is so simple that there's not much we can do to get a realistic answer to this question, but we could go so far as to consult our economic staff and ask for their opinions about the two possible outcomes:

*

*If the Democrats win, how much will the investment change?  (Suppose the answer is $d$ dollars.)


*If the Republicans win, how much will it change?  (Suppose the answer is $r$ dollars.)
The answers are numbers. To use them in the model, I will ask my staff to go through all the tickets in the box and on every "D" ticket to write "$d$ dollars" and on every "R" ticket to write "$r$ dollars."  Now we can model the uncertainty in the investment clearly and quantitatively: its post-election change in value is the same as receiving the amount of money written on a single ticket drawn randomly from this box.
This model helps us answer additional questions about the investment.  For instance, how uncertain should we be about the investment's value?  Although there are (simple) mathematical formulas for this uncertainty, we could reproduce their answers reasonably accurately just by using our model repeatedly--maybe a thousand times over--to see what kinds of outcomes actually occur and measuring their spread.  A tickets-in-a-box model gives us a way to reason quantitatively about uncertain outcomes.
Random variables
To obtain quantitative answers about uncertain or variable phenomena, we can adopt a ticket-in-a-box model and write numbers on the tickets.  This process of writing numbers has to follow only a single rule: it must be consistent. In the example, every Democratic ticket has to have "$d$ dollars" written on it--no exceptions--and every Republican ticket has to have "$r$ dollars" written on it.
A random variable is any consistent way to write numbers on tickets in a box.
(The mathematical notation for this is to give a name to the renumbering process, typically with a capital latin letter like $X$ or $Y$.  The identifying information written on the tickets is often named with little letters, typically $\omega$ (lower case Greek "omega").  The value associated by means of the random variable $X$ to the ticket $\omega$ is denoted $X(\omega)$.  In the example, then, we might say something like "$X$ is a random variable representing the change in the investment's value."  It would be fully specified by stating $X(\text{D})=d$ and $X(\text{R}) = r$.  In more complicated cases, the values of $X$ are given by more complicated descriptions and, often, by formulas.  For instance, the tickets might represent a year's worth of closing prices of a stock and the random variable $X$ might be the value at a particular time of some derivative on that stock, such as a put option. The option contract describes how $X$ is computed. Options traders use exactly this kind of model to price their products.)
Did you notice that such an $X$ is neither random nor a variable?  Neither is it "uncertain" or "unknown."  It is a definite assignment (of numbers to outcomes), something we can write down with full knowledge and complete certainty.  What is random is the process of drawing a ticket from the box; what is variable is the value on the ticket that might be drawn.
Notice, too, the clean separation of two different issues involved in evaluating the investment: I asked my economists to determine $X$ for me, but not to opine about the election outcome.  I will use other information (perhaps by calling in political consultants, astrologers, using a Ouija board, or whatever) to estimate the proportions of each of the "D" and "R" tickets to put in the box.

Afterward: about measurability
When the definition of random variable is accompanied with the caveat "measurable," what the definer has in mind is a generalization of the tickets-in-a-box model to situations with infinitely many possible outcomes.  (Technically, it is needed only with uncountably infinite outcomes or where irrational probabilities are involved, and even in the latter case can be avoided.)  With infinitely many outcomes it is difficult to say what the proportion of the total would be.  If there are infinitely many "D" tickets and infinitely many "R" tickets, what are their relative proportions?  We can't find out with a mere division of one infinity by another!
In these cases, we need a different way to specify the proportions.  A "measurable" set of tickets is any collection of tickets in the box for which their proportion can be defined. When this is done, the number we have been thinking of as a "proportion" is called the "probability."  (Not every collection of tickets need have a probability associated with it.)
In addition to satisfying the consistency requirement, a random variable $X$ has to allow us to compute probabilities that are associated with natural questions about the outcomes.  Specifically, we want assurance that questions of the form "what is the chance that the value $X(\omega)$ will lie between such-and-such ($a$) and such-and-such ($b$)?" will actually have mathematically well-defined answers, no matter what two values we give for the limits $a$ and $b$.  Such rewriting procedures are said to be "measurable." All random variables must be measurable, by definition.
A: A random variable, usually denoted X, is a variable where the outcome is uncertain. The observation of a particular outcome of this variable is called a realisation. More concretely, it is a function which maps a probability space into a measurable space, usually called a state space. Random variables are discrete (can take a number of distinct values) or continuous (can take an infinite number of values). 
Consider the random variable X which is the total obtained when rolling two dice. It can take any of the values 2-12 (with equal probability given fair dice) and the outcome is uncertain until the dice are rolled. 
A: From Wikipedia:

In mathematics (especially probability
  theory and statistics), a random
  variable (or stochastic variable) is
  (in general) a measurable function 
  that maps a probability space into a
  measurable space. Random variables
  mapping all possible outcomes of an
  event into the real numbers are
  frequently studied in elementary
  statistics and used in the sciences to
  make predictions based on data
  obtained from scientific experiments.
  In addition to scientific
  applications, random variables were
  developed for the analysis of games of
  chance and stochastic  events. The
  utility of random variables comes from
  their ability to capture only the
  mathematical properties necessary to
  answer probabilistic  questions.

From cnx.org:

A random variable is a function, which assigns unique numerical values to all possible
  outcomes of a random experiment under fixed conditions. A random variable is not a 
  variable but rather a function that maps events to numbers.

A: A random variable is a variable whose value depends on unknown events.  We can summarize the unknown events as "state", and then the random variable is a function of the state.
Example:  
Suppose we have three dice rolls ($D_{1}$,$D_{2}$,$D_{3}$).  Then the state $S=(D_{1},D_{2},D_{3})$. 


*

*One random variable $X$ is the number of 5s. This is:


$$ X=(D_{1}=5?)+(D_{2}=5?)+(D_{3}=5?)$$


*Another random variable $Y$ is the sum of the dice rolls. This is:


$$ Y=D_{1}+D_{2}+D_{3}  $$
A: The sample space may be a set of arbitrary elements, e.g. $\{\color{red} {\text{red}}, \color{green} {\text{green}}, \color{blue} {\text{blue}}\}$.
But this freedom of possible outcomes is difficult to work with. So someone invented the idea of working not with arbitrary elements, but only with real numbers.
To reach it, the first thing is to map such elements to real numbers, e.g.
\begin{aligned}
\color{red} {\text{red }} &\mapsto 6.72\\
\color{green} {\text{green}} &\mapsto -2\\
\color{blue} {\text{blue}} &\mapsto 19.5
\end{aligned}
And that mapping is called a random variable.
After choosing that mapping we get rid of problems how to operate with arbitrary things, because now we may do various calculations with numbers.
A: In the my non-math university studies, we were told that random variable is a map from values that variable can take to the probabilities. This allowed to draw the probability distributions

Recently, I have realized how different is that from what mathematicians do have in mind. It turns out that by the random variable they mean a simple function $X: \Omega \to \mathbb R,$ which takes an element of sample space $\Omega$ (aka outcome, ticket or individual, as explained above) and translates it into a real number $\mathbb R$ in range $(-\infty, \infty).$ That is, it was aptly noted above that it is not random and no variable at all. The randomness usually comes with probability measure $P,$ as part of measure space ($\Omega, P). ~P$ maps samples to $\mathbb R,$ similarly to random variable but this time range limited to $[0,1] $ and we can say that random variable translates $(\Omega, P)$ into $(\mathbb R, P),$ an thus, random variable is equipped with probability measure $P: \mathbb R \to [0,1]$ so that you can say for every $x \in \mathbb R$ what is the probability of its occurrence.
I do not know why do you need these kind of random variables and why cannot you sample the elements of R in the first place but it seems that translating samples to numeric values allows us to order the samples, draw the distribution and compute the expectation. I have got this idea reading A Measure Theory Tutorial (Measure Theory for Dummies) Might be mathematicians have better applications of random variable in mind, but I cannot find them in my superfluous study. The very same text suggests that you do not need converting samples into numbers always, particularly, to compute entropy for alphabet $\Omega$
$$H(\Omega) = \sum{P(\Omega_i) \ln (\Omega_i)}$$
integral does not need any real values of random variable.
A: Informally, a random variable is a way to assign a numerical code to each possible outcome.*
Example 1
I flip a coin. The set of possible outcomes (also called the "sample space") may be written as $\{H,T\}$. 
An example of a random variable $X$ might assign $X(H)=1$ and $X(T)=0$. That is, heads is "coded" as $1$ and tails is "coded" as $0$.
Example 2
I draw a card from a standard 52-card deck. The set of possible outcomes is $$\{A♠, K♠, \dots, 2♠, A♡, K♡, \dots, 2♡, A♢, K♢, \dots, 2♢, A♣, K♣, \dots, 2♣ \}.$$
In bridge, an ace is worth 4 high card points, a king 3, a queen 2, and a jack 1. Any other card is worth 0 points.
So we might let $Y$ be the corresponding random variable, where for example $Y\left(A♡ \right)=4$, $Y\left(J♣ \right)=1$, and $Y\left(7♠  \right)=0$.

What's the point of random variables? One simple answer is that abstract symbols like "$H$", "$T$" or "$A♠$" are sometimes difficult and troublesome to handle. So we instead translate them into numbers, which are easier to manipulate.
$$$$
*Formally a random variable is a function that maps each outcome (in the sample space) to a real number.
A: A random variable is a measurable function defined on a probability space:
If $\left(\Omega,\mathcal F, \mathbb P\right)$ is a probability space and $\left(S, \mathcal A \right)$ a measurable space, then any $\mathcal F/\mathcal A$-measurable function $Y: \Omega \to S$ is called a ($S$-valued) random variable.
Some authors use a less general definition in which $\left(S, \mathcal A \right) \equiv \left(\mathbb R, \mathop{\mathcal B}\left(\mathbb R\right) \right)$ is required, where $\mathop{\mathcal B}\left(\mathbb R\right)$ is the Borel $\sigma$-algebra on $\mathbb R$.
A: I was told this story:

A random variable can be compared with the holy roman empire:  The
  Holy Roman Empire was not holy, it was not roman, and it was not an
  empire.

In the same way, a Random Variable is neither random, nor a variable.  It is just a function.  (the story was told here:  source).
This is at least a quippy way to explain, which might help people remember!
A: Unlike a regular variable, a random variable may not be substituted for a single, unchanging value.  Rather statistical properties such as the distribution of the random variable may be stated.  The distribution is a function that provides the probability the variable will take on a given value, or fall within a range given certain parameters such as the mean or standard deviation.
Random variables may be classified as discrete if the distribution describes values from a countable set, such as the integers.  The other classification for a random variable is continuous and is used if the distribution covers values from an uncountable set such as the real numbers.
