Let me just set up some very standard notions. (I am working from the lecture series International Winter School on Gravity and Light 2015 on youtube)
$\textbf{Setup:}$ Given a d-dimensional topological manifold $(M,\mathcal{O}_M,\mathcal{A})$, here $M$ is the set, $\mathcal{O}_M$ the topology on $M$, and $\mathcal{A}$ is the associated atlas. Let $\rho:\mathbb{R}\to M$ be some path.
$\textbf{Definition:}$ The manifold is said to be smooth if the atlas $\mathcal{A}$ contains charts which are $C^\infty$ compatiable, that is two charts $(x,U_x),(y,U_y)$ are said to be $C^\infty$-compatible if $U_x\cap U_y=\emptyset$ or $$ y\circ x^{-1}: x(U_x\cap U_y)\to y(U_x\cap U_y),~\qquad\text{and}\qquad x\circ y^{-1}: y(U_x\cap U_y)\to x(U_x\cap U_y), $$ are $C^\infty$ (from $\mathbb{R}^d\to \mathbb{R}^d$).
In the youtube series, the professor introduces notion of the velocity of a curve $\rho$, he starts by assuming the curve is $C^1$. So it seems like before understanding what velocity is I need to know what it means for the curve $\rho$ to be $C^1$, he doesnt seem to explicitly define this notion, can anyone help?