Intuitive explanation of covariant, contravariant and Lie derivatives

The Lie derivative is a derivative of a vector field V along another vector field W. It is defined at a point p as follows: flow the point p along W for some time t and look at the value of V at this point. Then push this forward along the flow of W to a vector at p. Subtract $V_p$ from this, divide by t, and take the limit as $t \to 0$. So this is a measure of how V changes as it gets pushed around by the flow of W.

The covariant derivative is a derivative of a vector field V along a vector W. Unlike the Lie derivative, this does not come for free: we need a connection, which is a way of identifying tangent spaces. The reason we need this extra data is because if we wanted to take the directional derivative of V along the vector W how we do in Euclidean space, we would be taking something like $V_{p+tW} - V_p$, which is the difference of vectors living in different tangent spaces. If we have a metric, then we can impose reasonable conditions that give us a unique connection (the Levi-Civita connection).

I have no idea what a contravariant derivative is. I'd guess it has to do with applying a covariant derivative and lowering indices.