Let $f: M \mapsto N $ be a smooth map between differentiable manifolds. Then a vector field $X$ on $M$ is f-related to a vector field $Y$ in N if $$(f_*)_p (X_p) = Y_{f(p)} \ \ \forall p \in M$$
That is the derivative of the map $f$ at $p$ acting on a tangent vector at $p$ of $X$ is equal to a tangent vector at $Y$ at $f(p)$.
I'm struggling to understand the intuition behind this definition. How should one think about this? Is there a geometric interpretation?
Concretely, if $X$ is a vector field on $M$ and if $f : M \to N$ is an injective smooth map, then $\mathrm{d}f(X) = f_*X$ is a "vector field" on $f(M) \subset N$. $f(M)$ may not be a submanifold or anything, so it is not a "real" vector field. But if you can extend it in $N$, that is, if there exists a vector field $Y$ such that $f*X = Y$ at each $f(p) \in N$, then you say that $Y$ is $f-$related to $X$.
This definition is here to work with images of vector field by smooth maps as if they were "true" vector fields on $N$. Sometimes, one can not extend it (think of the tangent vector field of a self-crossing cuve in the plane).
Note that I asked $f$ to be injective, but one can just ask that if $f(p) = f(q)$, then $\mathrm{d}f(p)X_p = \mathrm{d}f(q)X_q$ so that the value of $f_*X$ at a point $z \in f(M)$ would not depend on the point in $f^{-1}(z)$.