How do I interpret the random variable $X_i$? "Suppose we have a random sample of n independent and identically distributed random variables $X_i$"
Hi, I am new to statistics. I am confused as to what $X_i$ represents. Are the $X_i$'s different variables? E.g. could $X_1$ be years of education, $X_2$ be age, etc? 
Thanks :) 
 A: No, because "Age" and "education" are not "identically distributed"
Consider instead a random sample of $n$ of those ages:
Age of first subject in sample = $X_1$
Age of second subject in sample = $X_2$
$\vdots$
Age of $n$th subject in sample = $X_n$ 
The material you quote is not well phrased and may be confusing if you don't already know what it's trying to say.
It seems as if it's referring to the entire collection as $X_i$, but in fact $X_i$ refers only to a particular one (the $i$th one). You can of course allow $i$ to take the values $1,2,...,n$ in turn, but that's not the same thing as being all of them at once (more typically you'd use bold or underline to indicate all of them together in a vector, say $\mathbf{X}=(X_1,X_2,...,X_n)^\prime$ ).
In a different situation to this, $X_1$ could represent a vector of ages, $X_2$ a vector of education-levels, and so on, and you'd refer to the individual values using $X_{1i}$, $X_{2i}$ and so on.
I have fudged/handwaved details of random variables a bit here but hopefully that gives a better sense of what that was talking about. You may like to read some of our questions and answers on random variables. whuber has a particularly simple and clear exposition here:
What is meant by a random variable?
