I've got a vague sense that all ODEs have an associated vector field, but I don't understand the exact relation between the two.I saw an answer here. It had several advanced math; fiber bundles, manifolds etcetera.
Is there a more rookie level explanation for this? Given a 2D vector field, say $$\vec{A}=\begin{bmatrix} P(x,y)\\ Q(x,y) \end{bmatrix}$$ How do you find its ODE? Or given a ODE, how am I supposed to find the corresponding Vector Field?
The ODE is the vector field. That is, the associated order 1 system is. In that case, the vector field is the same as the (vector valued) right side of the ODE. So in your example, one would associate the order 1 system $$ \dot x=P(x,y),\\ \dot y=Q(x,y). $$