Mutagens are those factors that cause mutation. Most sources list radiation and chemicals as the two types of mutagens. Some sources also add that there is a steady (background) amount of "natural" or "spontaneous" mutations. I haven't read enough about it to know whether this can be ascribed to a "natural" [terrestrial] amount of background radiation.
It is important to note that the implications of mutation - change in the original DNA base sequences of an organism - differ depending on whether the cell affected is a somatic or a germ cell. A mutation that occurs in a somatic, or body, cell will not affect heredity; the most notable outcome for a somatic mutation is cancer.
A mutation can be inherited if a) the DNA change has no significant impact on the viability of the cell, and the reproduction process, and b) it occurs in any of several types of germ cells - ie those cells that eventually give rise to the gamete egg or sperm.
Radiation in the ultraviolet wavelength can sometimes penetrate cells, and get absorbed by DNA, causing structural damage. Ionising radiation, which has a shorter wavelength than UV, is so called because it can knock electrons out of atoms, producing ions - which are more easily capable of taking part in chemical reactions. All organisms experience a small level of background radiation, which in origin can be either cosmic, solar or terrestrial (from either naturally occurring radioactive material or, more recently, human-originated).
At the DNA level, mutations can result in insertions of extra information (as base pairs), deletion of some pairs, or miscopying, such as transposition of sequences of base pairs.
Again, I wonder if "background" levels of ionising radiation are responsible for "background levels" of mutation. I would be interested to hear if any experiments have been done to establish causality: this could, for example, involve sufficiently shielding from background ratiation a sample of DNA-bearing organisms with a reasonably short generational span.