In Blumenhagen "Introduction to conformal field theory" book page no 12 it is proved that a holomorphic function $f(z)=z+\epsilon(z)$ gives a conformal transformation. My questions are:
what exactly the holomorphic function?
what is the main difference between holomorphic and anti-holomorphic?
Another question that can all holomorphic functions are able to give conformal transformation and is it is the only one to give such kind of transformation?
A physicist's answer would be that a holomorphic function is a function that depends on $z$ only, and not on $\bar{z}$.
For instance, the following expressions define holomorphic functions: $$ z^n \, , \sqrt{z} , \, e^z , \, \sin z , \, \frac{az+b}{cz+d} , \, \dots $$
On the other hand, $|z| = \sqrt{z \bar{z}}$ does not define a holomorphic function.
An anto-holomorphic function is a function that depends on $\bar{z}$ only, and not on $z$.
I repeat that this is an intuitive answer. For more rigorous statements, you can refer to the linked Wikipedia article.
As for your third question, it is not very clear to me what you are asking.