Jimi asked: I am a math guy, so sorry for the naivety. When I peruse the wikipedia I see many "variants" of quantum field theory... conformal quantum field theory, topological quantum field theory, axiomatic/constructive quantum field theory, algebraic quantum field theory, etc. Whether or not these are actually variants of something is unclear to me. I don't really have a specific question, but I was wondering if you guys could help me understand what these different things are and/or point me to somewhere to get a clearer picture.
They're variants, different kinds of quantum field theory, but they're not mutually exclusive. The different adjectives you mention separate quantum field theory to "pieces" in different ways. The different sorts of variants you mention are being used and studied by different people, the classification has different purposes, the degree of usefulness and validity is different for the different adjectives, and so on.
Conformal quantum field theory is a special subset of quantum field theories that differ by dynamics (the equations that govern the evolution in time), namely by the laws' respect for the conformal symmetry (essentially scaling: only the angles and/or length ratios, and not the absolute length of things, can be directly measured). Conformal field theories have local degrees of freedom and the forces are always long-range forces, which never decrease at infinity faster than a power law. They're omnipresent in both classification of quantum field theories - almost every quantum field theory becomes scale-invariant at long distances - and in the structure of string theory - conformal field theories control the behavior of the world sheets of strings (here, the CFT is meant to contain two-dimensional gravity but the latter carries no local degrees of freedom so it doesn't locally affect the dynamics) as well as boundary physics in the holographic AdS/CFT correspondence (here, CFTs on a boundary of an anti de Sitter spacetime are physically equivalent to a gravitational QFT/string theory defined in the bulk of the anti de Sitter space). Conformal field theories are the most important class among those you mentioned for the practicing physicists who ultimately want to talk about the empirical data but these theories are still very special; generic field theories they study (e.g. the Standard Model) aren't conformal.
Topological quantum field theory is one that contains no excitations that may propagate "in the bulk" of the spacetime so it is not appropriate to describe any waves we know in the real world. The characteristic quantity describing a spacetime configuration - the action - remains unchanged under any continuous changes of the fields and shapes. So only the qualitative, topological differences between the configurations matter. Topological quantum field theory (like Chern-Simons theory) is studied by the very mathematically oriented people and it's useful to classify knots in knot theory and other "combinatorial" things. They're the main reason behind Edward Witten's Fields medal etc.
Axiomatic or algebraic (and mostly also "constructive") quantum field theory isn't a subset of different "dynamical equations". Instead, it is another approach to define any quantum field theory via axioms etc. That's why it's a passion of mathematicians or extremely mathematically formally oriented physicists and one must add that according to almost all practicing particle physicists, they're obsolete and failed (related) approaches which really can't describe those quantum field theories that have become important. In particular, AQFTs of both types start with naive assumptions about the short-distance behavior of theories and aren't really compatible with renormalization and all the lessons physics has taught us about these things. Constructive QFTs are mainly tools to understand the relativistic invariance of a quantum field theory by a specific method.
Then there are many special quantum field theories, like the extremely important class of gauge theories etc. They have some dynamics including gauge fields: that's a classification according to the content. QFTs are often classified according to various symmetries (or their absence) which also constrain their dynamical laws: supersymmetric QFTs, gravitational QFTs based on general relativity, theories of supergravity which are QFTs that combine general relativity and supersymmetry, chiral QFTs which are left-right-asymmetric, relativistic QFTs (almost all QFTs that are being talked about in particle physics), lattice gauge theory (gauge theory where the spacetime is replaced by a discrete grid), and many others. Gauge theories may also be divided according to the fate of the gauge field to confining gauge theories, spontaneously broken QFTs, unbroken phases, and others. String field theory is a QFT with infinitely many fields which is designed to be physically equivalent to perturbative string theory in the same spacetime but it only works smoothly for open strings and only in the research of tachyon condensation, it has led to results that were not quite obtained by other general methods of string theory.
We also talk about effective quantum field theories which is an approach to interpret many (almost all) quantum field theories as an approximate theory to describe all phenomena at some distance scale (and all longer ones); one remains agnostic about the laws governing the short-distance physics. That's a different classification, one according to the interpretation. Effective field theories don't have to be predictive or consistent up to arbitrarily high energies; they may have a "cutoff energy" above which they break down.
It doesn't make much sense to spend too much time by learning dictionary definitions; one must actually learn some quantum field theory and then the relevance or irrelevance and meaning and mutual relationships between the "variants" become more clear. At any rate, it's not true that the classification into adjectives is as trivial as the list of colors, red, green, blue. The different adjectives look at the framework of quantum field theory from very different directions - symmetries that particular quantum field theories (defined with particular equations) respect; number of local excitations; ability to extend the theory to arbitrary length scales; ways to define (all of) them using a rigorous mathematical framework, and others.