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: 
If you have a *finite* presentation of a group $G$, then you can easily derive its abelianization $G^{ab}$ from that, by standard techniques: Basically, you create an integer matrix, with one column for each generator, and one row for each relation, noting in entry $a\_{ij}$ how often the generator $g\_j$ occurred in the relator $r\_i$. Then, compute the Smith normal form of this and you can read of the isomorphism type of $G^{ab}$.

If this happens to be non-trivial, you immediately have a proof that your group is non-trivial. Of course, the converse fails.
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).