Introduction
We know this is true: $Var(aX) = a^2Var(X)$
We also know this is true for independent variables: $$Var(X + Y) = Var(X) + Var(Y)$$ However, $Var(X + X) = Var(2X) = 4Var(X) \neq Var(X) + Var(X)$ So, what gives? First, some intuition. Our summation formula for independent variables was $2Var(X)$, whereas the real answer is $4Var(X)$.
Let's think about why this is. Let's say we have two variables X, and $Y = -X$. $$Var(X + Y) = Var(X + -X) = 0$$ When X and Y are positively correlated (in the first example), then when X + Y goes from really small (X and Y are both small) to really big (X and Y are both big). When X and Y are negatively correlated, then their sum is evened out, so there are no extremes. In fact, when Y = -X, it's always 0.
We also know this is true for independent variables: $$Var(X + Y) = Var(X) + Var(Y)$$ However, $Var(X + X) = Var(2X) = 4Var(X) \neq Var(X) + Var(X)$ So, what gives? First, some intuition. Our summation formula for independent variables was $2Var(X)$, whereas the real answer is $4Var(X)$.
Let's think about why this is. Let's say we have two variables X, and $Y = -X$. $$Var(X + Y) = Var(X + -X) = 0$$ When X and Y are positively correlated (in the first example), then when X + Y goes from really small (X and Y are both small) to really big (X and Y are both big). When X and Y are negatively correlated, then their sum is evened out, so there are no extremes. In fact, when Y = -X, it's always 0.
Covariance
Covariance is essentially how correlated two variables are.