- Imagine a QFT with some particle content. Some of these fields will be massless and some massive.
For simplicity, consider a massles scalar field and a massive scalar field with mass $M$.
If we are working at some energy $E\ll M$, we won't see the massive field (as happened with the Higgs before LHC, for example). This is the IR CFT.
Why IR? Because we are at low energy.
Why CFT? Well, the effective QFT that we are seeing under these conditions is, as I've said, made up of massles fields. These kind of theories are scale invariant, because you don't have any parameter in the theory that fixes a energy scale (a mass, or a coupling, or a characteristical length). Scale invariant theories have conformal symmetry, and are called CFT.
On the other hand, if now we take the energy to be $E\gg M$ we will se both fields, but now almost all of their energy will be momentum and their mass will be negligible. They will be behaving as two massless fields. That is the UV CFT.
Summing up, depending on the energy we are working on, the physics of our theory will change, and this is described by effective theories. It happens that at enough low energies and at enough high energies this effective theory is a CFT, IR and UV respectively.
- Roughly speaking, regularization is to introduce a cut-off so we can hide infinities.
Imagine a spins lattice. If we consider that the lattice is infinite, and we want to compute its energy, for example, we are going to obtain that it is infinite, just because we are summing a infinite number of contributions. So usually one says that the lattice has a volume $V$ and then it takes arbitrarily high values. This volume is called a regulator, and since it is related to long distances or low energies, it's called IR regulator.
On the other hand, if we want to consider that our lattice is a continuum, we will have problems too, because of having infinite degrees of freedom in a local region. So we introduce a separation length $\epsilon$ between the sites which goes to zero. This is the UV regulator, because it's related with short distances, or high energies.