A 3-manifold $N$ is called atoroidal if any incompressible torus is boundary parallel, i.e. can be isotoped into the boundary. To me, this definition assumes that $N$ has boundary, but I have read a number of times "an atoroidal closed 3-manifold" (eg in the hyperbolisation conjecture, see Hatcher's Classification of 3-manifolds).
If $N$ is closed, what does it mean that $N$ is atoroidal?
Depending on the exact definitions either (a) the manifold contains no embedded incompressible torus or (b) the fundamental group contains no $\mathbb{Z}^2$ subgroup. Some people assume that atoroidal includes (c) irreducible (and boundary irreducible).
On page nine of Hatcher's survey, he means (a).