The cross section is conceptually the area a nucleus presents to a target, having units of length squared. You can interpret this as roughly the relative probability of a nuclear intereaction occuring. Things with a high cross section have a much higher likelyhood than things with a low one.
Every kind of nuclear reaction has a cross section. An absorption cross section in most nuclear engineering roughly describes the probability of a neutron (or other particle) being absorbed by a nucleus.
An equation where this often shows up is in the 1D linear attenuation equation. The cross section (sigma) is usually referred to as a microscopic cross section as it is independent of material properties. The macroscopic cross section (capital sigma or sometimes mu) is the product of the number density A (units of m^-3) and the microscopic cross section (units of m^2) giving units of inverse length (m^-1). The macroscopic cross section is useful because it is a material property.
Say you have a beam of neutrons of intensity I0 incident on a thick slab of material. When the beam strikes the material, it will penetrate and some of the neutrons will be absorbed. The intensity of the beam at any point in the material is given as:
I(x) = I0*exp(-(macroscopic cross section)*x)
I(x) = beam intensity at position x
I0 = beam intensity at surface of target
x = distance in target.
Irradiation is the process of subjecting a material to some kind of radiation whether it be neutrons, beta particles, x-rays, microwaves, etc.