In dimension 2, there are two remarkable non-orientable closed manifolds, the projective plane (from synthetic geometry; has the fixed point property; algebraic compactification of the plane etc) and the Klein bottle (nowhere vanishing vector field; with immersions sold in your nearest nonorientable store). There is also a classification of all closed non-orientable surfaces, as connected sums of projective planes.

I am looking for examples of non-orientable 3 dimensional closed (compact, boundaryless) manifolds. Any with some special properties or arising from interesting geometrical problems? Is there a simple classification for them?

onlyassuming 'compact' important for you? If not, I think it would be good to make it 'non-orientable 3-dimensionalclosedmanifold'. Then, I think, people knowing much aboutThurston's geometrization conjecturecould, hopefully, relate your question to the current state of knowledge about geometrization. (For compact yet non-closed manifolds there is, I think, 'less' uniqueness in the geometric model structures, though much is understood for compact non-closed manifolds as well.) I am just suggesting you place your question 'squarely' intothe best-understood context of all. $\endgroup$