What specific gauges and gauge transformations are implied when one states that the order of such gauge groups are vital? Can this please be explained as simple as possible (:
Let’s take quantum chromodynamics (QCD), the current theory of quark-gluon interactions, as a specific example of a Yang-Mills theory. And let’s avoid equations!
There are three “colors” of quarks, usually called red, green, and blue. You can think of these colors as the QCD analog of electrical charge in QED. So there is a red quark field, a green quark field, and a blue quark field. (You have probably also heard of up quarks, down quarks, strange quarks, etc. This is related to weak interactions. Ignore this complication, and think about the three color fields for, say, an up quark.)
A gauge transformation mixes the three color fields in a linear way. For example, you might transform to a different set of quark fields which are various linear combinations of the original red, green, and blue fields. The theory is constructed to be insensitive to what “basis” (set of independent directions) you use for the “quark color space“. You can even choose a different basis for this quark color space at each point in spacetime!
In order to make it possible to choose a different color space basis at each point in spacetime, it turns out there must also be eight gluon fields with a particular way of transforming, which interact with the quarks and also with each other. Showing this, and finding how gluons must transform, requires some mathematics and not just words.
The transformations that mix the three quark color fields form a group known as $SU(3)$. It is similar to the group of rotations in 3D space, $SO(3)$, but in QCD it is mixing complex quantum fields rather than real spatial coordinates. Just as 3D rotations don’t commute, neither do linear transformations of quark color fields.
In short, the non-commuting gauge transformations of QCD are transformations of the abstract space of the quark color fields, and not transformations of spacetime. Choosing a gauge means, loosely, choosing which direction in color space will be called “red”, which “blue”, and which “green”, at each point.
Other Yang-Mills theories simply differ by the Lie group used to mix the matter fields. For example, you could have a set of fields mixing together under $SU(5)$, $SO(10)$, $Sp(27)$, $G_2$, $E_8$, etc. It is usually a good idea to take a math course on Lie groups and their “representations” to help you understand all the possible Lie groups and how to build Yang-Mills theories with the right set of particles that you want to have in them.
Many treatments of Yang-Mills theories will use 20th-century mathematical abstractions such as fiber bundles, differential forms, cohomology, etc. You may want to consider taking math courses on topology and differential geometry to have a good foundation in such mathematics, rather than just relying on an understanding of, say, differential equations. The additional insight will be worthwhile, especially if you want to understand how General Relativity relates to Yang-Mills theories.
If you have not yet encountered the ideas of abstract spaces, linear transformations, bases, etc., then take a course on linear algebra as soon as possible. Then follow it with a course on abstract algebra to introduce you to groups.