Question: Sorry for the catchy title. I want to understand, what should one have to do to disprove the Church-Turing thesis? Somewhere I read it's mathematically impossible to do it! Why?

Turing, Rosser etc used different terms to
differentiate between: "what can be computed" and "what can be
computed by a Turing machine".

Turing's 1939 definition regarding this is:
"We shall use the expression "computable function" to mean a function
calculable by a machine, and we let "effectively calculable" refer to
the intuitive idea without particular identification with any one of
these definitions".

So, the Church-Turing thesis can be stated as follows:
Every effectively calculable function is a computable function.

So again, how will the proof look like if one disproves this conjecture?

Answer: 
As far as I understand, the "impossibility" of proving or disproving the thesis is that there is no formal definition of "effectively calculable". Today, we take it to be precisely "computable by a Turing machine", but that rather begs the question. 

Models of computation that are strictly more powerful than a Turing machine have been studied, take a look at <http://en.wikipedia.org/wiki/Hypercomputation> for some examples. Or just take a Turing machine with an oracle for the Halting Problem for Turing Machines. Such a machine will have its own Halting Problem, but it can solve the original Halting Problem just fine. Of course, we have no such oracle, but there's nothing mathematically impossible about the idea.
The following papers from Selim Akl may be of interest and relevant to the discussion:

Akl, S.G., "Three counterexamples to dispel the myth of the universal computer", Parallel Processing Letters, Vol. 16, No. 3, September 2006, pp. 381 - 403.

Akl, S.G., "Even accelerating machines are not universal", International Journal of Unconventional Computing, Vol. 3, No. 2, 2007, pp. 105 - 121.

Nagy, M. and Akl, S.G., "Parallelism in quantum information processing defeats the Universal Computer", Parallel Processing Letters, Special Issue on Unconventional Computational Problems, Vol. 17, No. 3, September 2007, pp. 233 - 262.

Here is the abstract of the first one:

It is shown that the concept of a Universal Computer cannot be realized. Specifically, instances of a computable function F are exhibited that cannot be computed on any machine U that is capable of only a finite and fixed number of operations per step. This remains true even if the machine U is endowed with an infinite memory and the ability to communicate with the outside world while it is attempting to compute F. It also remains true if, in addition, U is given an indefinite amount of time to compute F. This result applies not only to idealized models of computation, such as the Turing Machine and the like, but also to all known general-purpose computers, including existing conventional computers (both sequential and parallel), as well as contemplated unconventional ones such as biological and quantum computers. Even accelerating machines (that is, machines that increase their speed at every step) cannot be universal.