What is the different semantic of existential and universal qualifier between classical and intuitionistic logic? Take the semantic definition of existential and universal qualifier outlined here
Is that valid for classical or intuitionistic logic? Or is it a general definition?

Take the following sentence:
"It exists a man that, if he owns an hat, then all mans own an hat"
Take $C(x)$ is the predicate:
"Man $x$ owns an hat"
than the sentence can be formally written as follow:
$\exists x(C(x) \rightarrow \forall xC(x)) $
How can that sentence be semantically interpreted using the initial definition of existential and universal qualifier?
 A: The semantic definitions you linked to in the first line of the question would be acceptable intuitionistically as well as classically, but an intuitionist would take "there exists an evaluation $\mu'$" to mean that one can actually exhibit an appropriate $\mu'$.  In other words, these definitions are intuitionistically acceptable in the context of an intuitionistic meta-theory.
From a philosophical viewpoint, one usually regards intuitionistic logic as differing from classical logic mainly in the interpretations of $\lor$ and $\exists$.  Intuitionism requires, for the assertion of $\phi\lor\psi$ or of $\exists x\,\phi(x)$, that one can (at least in principle) assert one of the disjuncts (in the case of $\lor$) or one specific instance (in the case of $\exists$). Because intuitionistic reasoning respects these requirements, proofs in intuitionistic systems often have far clearer computational content than proofs in similar classical systems.
A: One example is that in intuitionistic logic the usual De Morgan's laws do not hold (that is, only three out of four implications are true, e.g. see here), in particular
$$\neg(p \land q)\quad \text{ does not imply }\quad \neg p \lor \neg q.$$
Similarly
$$\neg \forall x. P(x) \quad \text{ does not imply } \quad \exists x. \neg P(x).$$
I hope it helps ;-)
Edit: Removed some false claims.
A: I think there's quite a nice explanation in the SEP article on constructive mathematics, in particular the section on the constructive interpretation of logic. It's best not to look at existential and universal quantifiers in isolation, but look at the interpretation as a whole, especially since disjunction and implication are also very important.
Since there are already good explanations around, I'll just give a quick example of when existential quantification is different in classical and intuitionistic logic:
Let $P$ be any proposition. For instance, take it to be some unproved conjecture eg the twin prime conjecture. In classical logic we can easily show that there is some number $n$ such that either $n$ is $0$ and $P$ is false or $n$ is $1$ and $P$ is true, by applying an instance of excluded middle $P \vee \neg P$. In intuitionistic logic you can't do this in general because you need to supply an actual witness for $n$. In particular you need to know whether $n$ is $0$ or $1$, which you can't do without either proving or disproving $P$.
