What does "a" at the last position mean? Can I get a concrete example? Here the notation means
$$T_{i_1,\dots, i_k}{}^{j_1\dots j_\ell} = T(\partial_{i_1},\dots,\partial_{i_k},dx^{j_1},\dots, dx^{j_\ell})$$
From what I understand the contraction is some sort of inner product, but the context is not entirely clear to me.
Here $a$ is literally just playing the role of $i_k$ and $j_\ell$. In other words, $$T_{i_1\dots i_{k-1} a}{}^{j_1\dots j_{\ell-1} a} = T(\partial_{i_1},\dots,\partial_{i_{k-1}},\partial_a,dx^{j_1},\dots, dx^{j_{\ell-1}},dx^a).$$ Note that $\partial_a$ and $dx^a$ make sense because $a$ ranges from $1$ to $n$.