Question: This maybe a very general question.

If we have a group given by its presentation only, what kind of properties could be proven about it?

I know examples about non-amenability of some Burnside groups.

What kind of examples are there in literature where one proves some property "just" from a presentation?

Answer: 
*Almost nothing* can **reliably** be said about a group just from a presentation **in finite time**. (In fact, the abelianisation is just about the only thing one can reliably compute.) Most strikingly, there is no algorithm to recognise whether a given presentation represents the trivial group. More generally, one cannot in general solve 'the word problem' - ie, there is no algorithm to determine whether a given element is non-trivial. See [Chuck Miller's survey article](http://www.ms.unimelb.edu.au/~cfm/papers/paperpdfs/msri_survey.all.pdf) for details.

(**Update.** I inserted the word 'reliably' above in deference to Joel David Hamkins' fair comment. (**Update 2.** I then inserted the phrase 'in finite time' to be strictly correct, in an effort to head off further argument.) It is true that, in many special cases, there is information that can be read off from a specific presentation. This is more or less the topic of combinatorial group theory! But I want to emphasise that you can do nothing with an arbitrary presentation.)

On the other hand, there is a growing realisation that, surprisingly, if one is given a solution to the word problem (by an oracle, say) then one can compute quite a lot of information. Daniel Groves and I proved that, in these circumstances, one can determine whether the group in question is free. [Nicholas Touikan](http://arxiv.org/abs/0906.3902) generalised this to show that one cam compute the [Grushko decomposition](http://en.wikipedia.org/wiki/Grushko_theorem#Grushko_decomposition_theorem).
Given a group $\Gamma$ which you are interested in, given by a presentation, and an easy group $G$ which you understand (a dihedral group or a symmetric group, maybe), you can often use the presentation to determine whether or not there exists a surjective homomorphism $\Gamma\to G$, and maybe you can even count them. This is a strong tool for showing that two given presentations give rise to different groups, and for showing that your group must be "at least as complicated" as $G$. For instance, Tietze originally proved in 1908 that the trefoil is knotted by exhibiting a surjection from a Wirtinger presentation of the fundamental group of its complement onto the symmetric group $S\_3$, while the fundamental group of the unknot is abelian and so can admit no such homomorphism.