To make the connection to the Lie derivative, let $t \mapsto \Phi^X_t$ be the 1-parameter group of diffeomorphisms (or flow) generated by the vector field $ X $. The differential $ d\Phi^X_t $ of each diffeomorphism maps the vector field Y to a new vector field $ \mathrm{d}\Phi^X_{t}(Y)$. (http://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields)
Not sure why $ d\Phi^X_t $ would map vector field $Y$ to a new vector field. Can anyone explain this? Also, what is the difference between $ \mathrm{d}\Phi^X_{t}$ and $ d\Phi^X_t $?