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: 
Quite a lot, I believe (although `a lot' is subjective).

A neat example of a property which can be read immediately off of a (finite) presentation $\langle X; R \rangle$ is the Deficiency of said presentation. This is defined to be $|X|-|R|$. Now, this contain some intriguing properties. For example, every group of deficiency greater than 2 is large (it has a finite-index subgroup which contains a homomorphism onto a non-abelian free group), while every group of deficiency 1 is infinite and if a deficiency 1 presentation has a relator which is a proper power then this group too is large.

Also, the word, conjugacy, etc. problems for presentations is an intriguing topic.
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).