What do they mean when they say "random variable"? 
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:


$$ Y=D_{1}+D_{2}+D_{3}  $$
