Intuitive

Conformal field theory is either a classical field theory or quantum field theory with only massless particles. In such theories, there is no mass scale and hence the theory is scale invariant, which means it looks the same no matter from how far away we look at it.

Concrete

Abstract

Why is it interesting?

The modern way of understanding relativistic quantum field theories (QFTs) is through renormalization group flows away from conformal field theories (CFTs). In the parameter space of all QFTs, CFTs arise as fixed points with enhanced scale and conformal symmetry. The very ambitions programme of understanding all QFTs thus is intimately related to the classification of all consistent CFTs.https://iopscience.iop.org/article/10.1088/1361-6382/aa8003/pdf

It is an old idea in particle physics that, in some sense, at sufficiently high energies the masses of the elementary particles should become unimportant. In recent years this somewhat vague hope has acquired a more definite form in the theory of scale transformations, or dilatations.

Aspects of Symmetry: Selected Erice Lectures by Sidney Coleman

In particle physics, one longstanding hope has been that at high energies, particle masses can be neglected, so that the physics would become scale invariant. It turns out that in a local field theory, it is true, more or less in general, that scale invariance typically leads to conformal invariance. (This is because the violation of scale invariance and conformal invariance are both determined by the trace $T^\mu_\mu$ of the energy momentum tensor.)

page 621 Einstein Gravity in a Nutshell - A. Zee

It seems that all QFTs can be viewed as points along an Renormalization Flow (or RG flow, this is the name we give to the zooming process) from a ‘UV’ CFT to another ‘IR’ CFT. Renormalization flows occur when we deform the UV CFT, breaking its conformal symmetry. […] Well-defined QFTs can be viewed as either CFTs or as RG flows between CFTs. We can remove the UV cutoff from a QFT (send it to infinite energy or zero length) if it can be interpreted as an RG flow from the vicinity of a CFT fixed point. So studying the space of CFTs basically amounts to studying the space of all well-defined QFTs.

Lectures on AdS/CFT from the Bottom Up by Jared Kaplan

See also: https://www.quantamagazine.org/using-the-bootstrap-physicists-uncover-geometry-of-theory-space-20170223 and Critical Exponents