Let $V$ be a $n$ dimensional vector space over $\mathbb R$ and $k,p,q\in\mathbb N$.
A $k$-tensor is a $k$-multilinear functional on $V$, that is a map \begin{align} f:&& V^k&\longrightarrow\mathbb R \\ &&(v_1,\dotsc,v_k)&\longmapsto f(v_1,\dotsc,v_k) \end{align} such that $f$ is linear with respect to each variable.
A tensor of type $(p,q)$ is a functional \begin{align} T:&& (V^*)^p\times V^q&\longrightarrow\mathbb R \\&&(f_1,\dotsc,f_p),(v_{1},\dotsc,v_q)&\longmapsto T\big(f_1,\dotsc,f_p,v_1,\dotsc,v_q\big) \end{align} which is linear with respect to each variable.
I have a little confusion here: what is the difference in meaning between these two concepts? Do they mean the same or different thing? For which reason do people distinguish these concepts? When one say "a tensor" in mathematics, which kind of tensor is usually being referred to? And when one writes a math thesis which needs to use a tensor concept, like differential geometry or applying math in relativity theory, should the concept of tensor of type $(p,q)$ be abandoned and just focus on the $k$-tensor?