Yes, the vorticity. A flow is irrotational if and only if vorticity (the curl of velocity) is zero at all points.
If the fluid is assumed to be incompressible, irrotational flows are interesting because the momentum equation is reduced to the Laplace equation for the velocity potential. Laplace equation is linear, so it admits a superposition of elementary solutions. This the basis of what is called the potential flow theory:
Potential flow - Wikipedia
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