Two maps $f : X \to Z$ and $g : Y \to Z$ of manifolds are transversal roughly if the images of $X$ and $Y$ in $Z$ do not “touch tangentially”.
Two maps $f : X \to Z$ and $g : Y \to Z$ of smooth manifolds are transversal if for all point $x \in X$ and $y \in Y$ with $f(x) = z = g(y)$ the differentials of $f$ and $g$ in these points span the entire tangent space at $z$ in the sense that
Note that this is not required to be a direct sum. Also, if $f$ (say) is a submersion, then it is transversal to all $g$.
In particular, $f$ or $g$ may be inclusions of (possibly immersed) submanifolds in which case we talk about the transversality of submanifolds.
Various constructions involving pullbacks of manifolds work as expected only for pullbacks involving transversal maps.
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, $T(X \times_Z Y) = T X \times_{T Z} T Y$. (However, the pullback may exist and be preserved without transversality; for example if $X$ and $Y$ are both abstract points, $Z$ is not a point, and the maps $X, Y \to Z$ are equal as concrete points of $Z$.)
This is to be regarded as the dual of the possibly more familiar statement that various constructions involving quotients only work as expected for free actions.
Both of these “problems” are solved by passing from the ordinary $1$-category of manifolds to a suitable higher category of generalized smooth spaces.
More precisely:
the problem with the pushouts (quotients) is resolved by passing to stacks and smooth infinity-stacks.
the problem with the pullbacks is resolved by passing to derived stacks. Concretely for the case of manifolds this is discussed at derived smooth manifold.
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