Of the properties of mathematics, as a language, the most peculiar one is that by playing formal games with an input mathematical text, one can get an output text which seemingly carries new knowledge. The basic examples are furnished by scientific or technological calculations: general laws plus initial conditions produce predictions, often only after time-consuming and computer-aided work. One can say that the input contains an implicit knowledge which is thereby made explicit. One could try to find a parallel in the humanities by comparing this to hermeneutics: the art of finding hidden meanings of sacred texts. Legal discourse, too, has some common traits with scientific discourse. In the course of history, the modern language of science slowly emerged from these two archaic activities, and it still owes a lot to them, especially in the more descriptive and less mathematicised domains.
The goal of a definition is to introduce a mathematical object. The goal of a theorem is to state some of its properties, or interrelations between various objects. The goal of a proof is to make such a statement convincing by presenting a reasoning subdivided into small steps each of which is justified as an "elementary" convincing argument.
To put it simply, we first explain what we are talking about, and then explain why what we are saying is true (pace Bertrand Russell).
A proof only becomes a proof after the social act of "accepting it as a proof".