How is the action of a Lie group element on a tangent vector defined? I am trying to understand the concept of a left-invariant vector field, much as in this question here. I am not clear on what is meant by "derivative of left-multiplication by $g$". How is this derivative defined? How is the action of an element of the group on an element of the tangent space defined?

If you fix an element $g\in G$ in your Lie group, then it defines a smooth (by definition) map $l_g\colon G\to G$ given by $l_g(h)=gh$, for any $h\in G$. Since $l_g$ is just a smooth map between two smooth manifolds, you can take its derivative at any point, say, at $h\in G$. You get $d_h(l_g)\colon T_hG\to T_{gh}G$.
