I've had a look for intuitive explanations of the variance of an RV (e.g. Intuitive explanation of variance and moment in Probability) but unfortunately for me, I still don't feel comfortable with the concept. Why would you opt to use variance over the standard deviation (which usefully is in the same units as the expectation)?
Also, if the expectation,
$E(X) = \Sigma_{i=0}^n i P(X = i)$,
what is $E(X^2)$? Is it simply
$E(X^2) = \Sigma_{i=0}^n i P(X^2 = i)$?