An irrotational vector field is a vector field where curl is equal to zero everywhere. If the domain is simply connected (there are no discontinuities), the vector field will be conservative or equal to the gradient of a function (that is, it will have a scalar potential).
Similarly, an incompressible vector field (also known as a solenoidal vector field) is one in which divergence is equal to zero everywhere. Such a vector field will have a vector potential (it will be equal to the curl of some function). By Kelvin–Stokes theorem, a flux integral in an incompressible vector field will depend only on the boundary, and any closed flux integral will have a value of zero.
By Helmholtz's theorem, any vector field can be written as the sum of a gradient and a curl (or, in other words, an irrotational and incompressible vector field).