What can be said about a group from its presentation? 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?
 A: 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.
A: 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. 
A: 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 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.
A: 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.
A: 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 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 generalised this to show that one cam compute the Grushko decomposition.
