Let $M$ be a manifold and $\varphi:U_0\to U$ be a parametrization. For every $x=\varphi(u)\in U$, although the point $y$ is in $\mathbb R^n$, it needs only $m$ numbers to have its position determined, we say, the $m$ coordinates of $x\in U_0$ which we call parameters.
Afterwards the author of the book I'm studying say $\{\frac{\partial \varphi(u)}{\partial u_1},\ldots,\frac{\partial\varphi(u)}{\partial u_m}\}$ is a basis of the tangent vector space $T_xM$.
So I'm a little confused with the notation. In the first paragraph the $u_1,\ldots, u_m$ are numbers and $u$ is a fixed point in $U_0$. On the other hand, in the second one they are vectors.
After all, what are they? could anyone give me a more concrete example?
I need help, the author didn't say anything else about this discussion.