- The prime decomposition theorem is slightly different for non-orientable 3-manifolds. There is a version decomposing along 2-spheres described in Hempel’s book, but there is a subtle flaw in the statement corrected by Bruce Trace. A possible more natural decomposition is along projective planes and 2-spheres, but here the prime summands may be orbifolds with isolated orbifold points which are the cone over a projective plane. However I’m not sure if this is written down somewhere.
As pointed out by Moishe Kohan, the geometric decomposition for non-orientable 3-manifolds is described in the last section of Scott’s “The Geometries of 3-manifolds”. Unfortunately this doesn’t seem to have been proved in full generality yet.
Classes of examples of non-orientable 3-manifolds may be made by taking mapping tori either of non-orientable surfaces or of non-orientable mapping classes of orientable surfaces. Note that if $M$ is closed, aspherical, and non-orientable, then $b_1(M)>0$ and $M$ is Haken. Another simple way to construct non-orientable manifolds is to take a complement of a tubular neighborhood of a knot and attach the boundary to a torus by a double cover where the meridian wraps twice and the longitude once.
There are no closed non-orientable 3-manifolds modeled on spherical, Nil, or $\widetilde{SL_2(\mathbb{R})}$ geometries. Manifolds modeled on these geometries are Seifert fibered with non-triviald Seifert bundle. If a Seifert fibered manifold is non-orientable, then the Euler class of the Seifert bundle of the two-fold orientable cover is zero (since there is an orientation-reversing involution either reversing the orientation of the fibers or the base), so the manifold cannot be modeled on one of these geometries.