What's the difference between perturbative QCD, non-perturbative QCD, and gauge theory QCD?

Most quantum field theories cannot be solved exactly, so physicists have developed approximation schemes.

The perturbative approach to quantum field theory relies on expansions in the interaction coupling constant. For example, in QED one expands quantities in powers of the fine-structure constant $\alpha=e^2/\hbar c$, which measures how strongly electrons couple to photons. Since this is a small number (about 1/137), calculating the first few terms of order $\alpha$, $\alpha^2$, $\alpha^3$, etc. tends to often give a good approximation to whatever quantity you are calculating. (Unfortunately the whole series doesn’t converge, but that’s another story.) Feynman diagrams are a pictorial way of representing these complicated perturbative calculations. Things like the anomalous magnetic moment of the electron have been calculated to one part in a trillion using this approach.

However, the bad news is that some phenomena, like quark confinement in QCD, are completely overlooked by doing a perturbation expansion. And some interactions, like the strong interaction, have a coupling constant which isn't particularly small. So physicists have developed non-perturbative approaches to quantum field theories as well.

A common non-perturbative approach, frequently used for QCD, is to formulate the theory on a spacetime lattice of finite extent rather than allowing spacetime to be continuous. This reduces the number of degrees of freedom in the theory to a finite number such that states of the theory can be represented in a computer.

As for gauge theory, QED and QCD are specific examples of gauge theories. There is not a variant of QCD called Gauge Theory QCD. Gauge theories have non-spacetime symmetries in addition to the usual Poincaré invariance under spacetime transformations. For example, in QCD an $SU(3)$ symmetry expresses the invariance of the theory under linearly mixing the quark colors red, green, and blue. The gauge symmetry basically says, “the three quark colors are equivalent” in the same way that rotational invariance says “$x$, $y$, and $z$ are equivalent”.

In physics, once one has a theory, a mathematical model, one starts using it to compute measurable quantities to test against data, or predict the result of experiments.

QCD is a gauge theory, and it is part of the standard model which is the theory for particle physics, SU(3)xSU(2)xU(1).

In physics, a gauge theory is a type of field theory in which the Lagrangian does not change (is invariant) under certain Lie groups of local transformations.

The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory.

The reason these particular groups are used for the standard model is because they fit the symmetries discovered experimentally in the data of elementary particles.

To be able to calculate quantities with this model, different tools are used. For the SU(2) and the U(1) groups things are simpler. The Feynman diagram recipe which is a pictorial representation of the perturbative expansion of the (assumed) true solution, converges, because the series terms (which is what the perturbative part means) converge, and the next term in the series is smaller than the previous. (There are corrections that enter when one wants to include all the possible Feynman diagrams, but that is a story to learn when studying the specific field theories).

In the case of QCD, the coupling constant that would characterize a series expansion is 1, so it is only qualitatively one can use Feynman diagrams. The series diverges. So perturbative QCD is only for illustration purposes, not calculation, as where strong interactions take place there is an infinity of quarks antiquarks and gluons that have to be considered, each vertex with a coupling of 1.

The most successful calculational method for strong interactions is QCD on the lattice, which has given calculations for the masses of complex particles.

Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum QCD is recovered.

So QCD is a gauge theory, perturbative Feynman diagrams are only for illustration purposes, and there do not exist perturbative methods for calculating crossections, etc. for strong interactions which are due to QCD. Other mathematical tools are used, called non-perturbative in contrast.