Connection between (1,1) tensors and maps of vector fields

I'm taking a course in differential geometry and my professor used some notation that confused me.

Let $X$ be a vector field and let $J$ be a $(1,1)$ tensor (defined as being an element of $V\otimes V^*$). My question is this: what does the notation $J(X)$ mean?

I know that there have been lots of questions on this site relating $(1,1)$ tensors to linear maps and the answers to the questions have only somewhat helped. I understand that there is a map $\langle v,v^* \rangle \to \mathbb{R}$ taking $ v\otimes v^* \mapsto \sum_{i} v_iv^*_i$ and that this can somehow also be viewed as a map taking derivations to derivations via $X|_p \mapsto \langle X|_p \cdot \rangle$ but I'm not entirely sure if I'm understanding this correctly.

Any help would be greatly appreciated.