One can derive an equation that gives the error in an image on the image plane as a function of the incident (imperfect) wavefront. 1st order aberrations vary as h (where h is the distance from the centre of the lens), 3rd order aberrations vary as h^3 and so on. The reason there is no 2nd, 4th, 6th.. order aberrations (i.e. no h^2, h^4, h^6...dependances) is because of the cylindrical symmetry of the lens.
There is only 1 first order aberration and it is called defocus.
There are 5 third order aberrations;
Curvature of Field
The origin of these aberrations (i.e. why they appear) can be found here;