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: 
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.
Continuing the idea that working with bare presentations is "hard", automatic groups sometimes can give you a handle, if you're trying to study a specific group . [Automatic groups](http://en.wikipedia.org/wiki/Automatic_group) are groups with finite state machines that can, essentially, solve the word problem for that group. 

If you're studying a particular group and are lucky, a procedure such as Knuth-Bendix can compute an automatic structure for you. Then lots of hard computations become easy (e.g. the order of the group).

Magma has some of these algorithms implemented, see [this Magma documentation page](http://magma.maths.usyd.edu.au/magma/htmlhelp/text869.htm).