The following is an excerpt from Dynamical Systems by Shlomo Sternberg:
By a transversal, $L$, to the vector field $V$ we mean a surface of codimension one which is nowhere tangent to $V$ . In the plane, this means that $L$ is a curve.
In particular, the vector field, $V$ does not vanish at any point of $L$. If $V (A) \not= 0$, we can always find a transversal to $V$ passing through $A$: Simply choose a subspace of codimension one of the tangent space at $A$ which does not contain $V (A)$, and then choose a surface tangent to this subspace at $ A$. At all points sufficiently near to $A$ the vector field $V$ will not be tangent to this surface on account of continuity.
I don't understand the part "If $V (A) \not= 0$, we can always find a transversal to $V$ passing through $A$: Simply choose a subspace of codimension one of the tangent space at $A$ which does not contain $V (A)$, and then choose a surface tangent to this subspace at $ A$." Can any one come up with a picture showing how this is done in ${\Bbb R}^2$ and ${\Bbb R}^3$?
In ${\Bbb R}^2$ and ${\Bbb R}^3$ it's very easy, since you don't have to make the distinction between the manifold and its tangent space. Just take $L$ to be the line (in ${\Bbb R}^2$) or the plane (in ${\Bbb R}^3$) which passes through the point $A$ and has $V(A)$ as its normal vector.