I read John Hempel's paper "3-manifolds viewed from the curve complex". In the proof of Theorem 3.2, he calls annuli in the boundary of a solid torus essential. (see picture below from his article).
All definitions that I find of essential annulus requiring that the annulus is properly embedded. Hence, the definitions do not make sense.
Let $M$ be a 3-manifold with boundary. Let $A$ be an annulus in $\partial M$. What does it for $A$ mean to be essential?