The Lie derivatives quantifies the evolution of a tensor field along the flow generated by a vector field. Algebraically, the Lie derivative $L_X$ along $X$ satisfies the following four properties:

1. $L_X f = Xf$ on functions
2. $L_X (S \otimes T) = (L_X S) \otimes T + S \otimes (L_X T)$
3. $[L_X, d] = 0$
4. $L_X (T(Y_1, \ldots, Y_n)) = (L_X T)(Y_1, \ldots, Y_n) + T(L_X Y_1, \ldots, Y_n) + \cdots + T(Y_1, \ldots, L_X Y_n)$

These axioms imply that, for example, $L_X Y = [X,Y]$ for $Y$ a vector field.

On forms there is also the Cartan 'magic formula' $L_X = d \iota_X + \iota_X d$.

Recall (p. 24 of DG notes) that $L_X = \nabla_X + e(\omega^i) e^*(\nabla_{e_j} X)$.