Which is the definition of a conformal field theory?

"The" definition does not exist. There are many (related but not completely equivalent) definitions, depending on the perspective. Some of them are mentioned in the review here.

Which are the physical prerequisites one would need to start studying conformal field theories? (i.e Does one need to know supersymmetry? Does one need non-perturbative effects such as instantons etc?)

Instantons cannot arise in CFT. Supersymmetry can, but much of CFT can be understood without it. A knowledge of the meaning of free quantum fields and the associated machinery is essential, and understanding the Wightman axioms is helpful.

Which are the mathematical prerequisites one would need to start studying conformal field theories? (i.e how much complex analysis should one know? Does one need the theory of Riemann Surfaces? Does one need algebraic topology or algebraic geometry? And how much?)

You need a good acquaintance with complex analysis (Laurent series) and with semisimple Lie algebras and their representations. If you are only interested in CFT on the torus, neither Riemann surfaces nor algebraic geometry is needed. if you are interested in CFT at genus $>0$ (needed, e.g., for string theory) you need some basics in both areas, and on category theory.

Which are the best/most common books, or review articles, for a gentle introduction on the topic, at second/third year graduate level?

For the case of 2 dimensions, I recommend the review article by Fuchs, the lecture notes by Ginsparg, and the books by Kac (Vertex algebras for beginners) and di Francesco (Conformal field theory). In higher dimensions I recommend Rychkov's Lecture notes on CFT.

Do CFT models have an application in real world (already experimentally tested) physics? (Also outside the high energy framework, maybe in condensed matter, etc.)

At a critical point, most thermodynamic systems are described by a corresponding CFT. See, e.g., Lecture 5 in the book by Ginsparg mentioned above. The dynamics of the quantum Hall effect is also governed by CFT.