Two maps and of manifolds are transversal roughly if the images of and in do not “touch tangentially”.
Two maps and of smooth manifolds are transversal if for all point and with the differentials of and in these points span the entire tangent space at in the sense that
im(d f) + im(d g) \simeq T_z Z \,.
Note that this is not required to be a direct sum. Also, if (say) is a submersion, then it is transversal to all .
In particular, or may be inclusions of (possibly immersed) submanifolds in which case we talk about the transversality of submanifolds.
For example, two maps with a common target are transversal only if their pullback exists and is preserved by the tangent bundle functor; that is, . (However, the pullback may exist and be preserved without transversality; for example if and are both abstract points, is not a point, and the maps are equal as concrete points of .)
T. Bröcker, K. Jänich, C. B. Thomas, M. J. Thomas, Introduction to differentiable topology, 1982 (translated from German 1973 edition; ∃ also 1990 German 2nd edition)
Morris W. Hirsch, Differential topology, Springer GTM 33, gBooks