I am a little confused about integration of curves in manifolds. An element of a tangent bundle is a vector field called $X$, made of a set of vectors tangent to a point $p$. How can we find the curve passing through the point $p$ as an integral curve of vector field? On the one hand, $X$ gives us $\dot g =X(g)$ with $g(0)=p$, but isn't the integral curve of $X$ only related to the point $p$, because $X$ is only related to the point $p$, but $g$ passes through many points?
The Frobenius theorem talks about local integrability and collections of vector fields, which may solve this, but I don't understand in what way. Since I'm only starting out in this topic, I don't really understand what is the difference between this stuff and typical diff eqs stuff is.
Thanks for any help you can offer, including references or basic definitions that I may have confused.