Analytic is a term that describes a complex function that is "differentiable" at and in the neighborhood (region of non-zero area) surrounding some point. A function that is analytic over the entire complex plane is called entire.
But this is an empty definition unless we clarify what it means for a complex function to be differentiable. Differentiability of a complex function f(z = x+iy) is a very strict criterion. f(z) is defined to be differentiable at a point z0 iff the limit as Δz goes to 0 of (f(z0+Δz) - f(z0))/Δz exists and is independent of how Δz goes to 0. For example, the limit must be the same for both the case when Δz = Δx and the case when Δz = iΔy. More generally the limit must be the same for all 360 degrees worth of directions of Δz in the complex plane.
The mathematical necessary conditions for a function f(z) to be differentiable at a point are called the Cauchy-Riemann equations, and are very easy to prove. It can be proven that given any analytic function, its first derivative is analytic as well, implying that every analytic function has derivatives of all orders in the region in which it's analytic. This statement is not true for differentiable functions of real variables. For instance f(x) = x|x| is differentiable at x = 0, but its derivative is not.
Analytic functions have wonderful mathematical properties, and I am unaware of interest in or work on nonanalytic function theory.