What is the significance of the `3d_(x^2-y^2)` atomic orbital?

Significance? Well, it corresponds to one of the magnetic quantum numbers in the set `m_l = {-2, -1, 0, +1, +2}`, and has `m_l = -2` by convention.

The `3d_(x^2 - y^2)` orbital is the one lying along the axes:

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COMMON BONDING CASES

Because of that, it is often used to `sigma` (sigma) bond with surrounding ligands, particularly in a transition metal complex. You'll see a strong contribution from this orbital most often in...

square planar complexes:


and octahedral complexes:

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ENERGIES

In an atom, it is degenerate with the other `"four"` `3d` orbitals, the `d_(z^2)`, `d_(xy)`, `d_(xz)`, and `d_(yz)`.

In a complex, the `d` orbitals split in accordance with their symmetry (horizontal correspondence) and energy (vertical separations) with respect to the other `d` orbitals:


(For the octahedral complex, the symmetries from top to bottom were `E_g` and `T_(2g)`, and for the square planar complex, the symmetries from top to bottom are `B_(1g)`, `B_(2g)`, `A_(1g)`, and `E_g`.)