The laws of electrodynamics, especially that there is a conserved coulomb charge, can be explained compactly by the fact that the laws of nature are invariant with respect to an internal (one that has nothing to do with the "familiar" four dimensional spacetime) rotation. Shortly, one invariance = one conserved charge.
In order to explain other laws of conservation observed in high energy physics one has introduced several other internal rotations (for simplicity you may think of spatial rotations, but it's slightly more complicated than that). Thus one gets several other conserved charges (e.g. the quarks). Likewise one gets several "force fields" (e.g. the gluon fields) analogous to the electromagnetic field.
But whereas the Maxwell equations of electrodynamics are linear, the Yang-Mills equations become non-linear as a consequence of the fact that two subsequent rotations don't give the same result when their order is exchanged. You can convince yourself of the latter with an arbitrary object in your hands, by applying two perpendicular 90 degree rotations. Nonlinearity adds much complexity and this is the reason why accurate predictions are comparatively difficult in YM theory.