Intuitively, Gromov-Witten theory makes perfect sense. Via Poincare duality, we look at the cohomology classes $\gamma_1, \ldots, \gamma_n$ corresponding to geometric cycles $Z_i$ on a target space $X$, pull them back and then the integral $$\langle \gamma_1 \cdots \gamma_n\rangle=\int_{\overline{\mathcal{M}}_{g,n}(X,\beta)^{vir}}ev_1^*\gamma_1 \smile \cdots \smile ev_n^*\gamma_n$$ should count the number of curves whose intersection with the given cycles is non-empty.

However, we also have the ψ-classes (or "gravitational descendants") arising from the moduli space $\overline{\mathcal{M}}_{g,n}$ which are the chern classes of the $i$-th cotangent line bundle to a given $(C, x_1, \ldots, x_n) \in \overline{\mathcal{M}}_{g,n}$.

So what, geometrically, do these represent? The fact that they arise from $\overline{\mathcal{M}}_{g,n}$ means that the inclusion of a ψ-class places restriction on the geometry of the curves which we count; that much is clear. What is this restriction?

The reason that I am curious is that I am trying to evaluate the GW-invariants corresponding to maps which have components collapsing to an A_{1} singularity (i.e. a $B\mathbb{Z}/2$), but such that not all of the curve collapses. It has been mentioned in passing that including a ψ-class could help with this, and while the little I understand makes this sound plausible, I don't exactly see why.

So what are ψ-classes? Can I use them to split my curve up into parts so that a fixed component lands on my stacky point, while the rest of it does whatever else curves do?