When a particle of matter meets a particle of antimatter, they annihilate transforming themselves in energy (gamma ray).
According to the theory of the Big Bang, in the early universe, matter and antimatter were supposed to be present in equal proportions and consequently they should have caused an immediate annihilation process that had to obliterate the entire newly formed universe.
Since this does not correspond to the reality we observe, it is believed that a slight imbalance in favor of matter (known as the violation of CP symmetry) allowed it not to be completely annihilated, enabling the formation of the universe in which we live through the process of baryogenesis.
But what "helped" the matter winning the fight against the antimatter?
There are some theories that explains also other phenomenons in the universe, but they are still not totally confirmed, in fact there are some experiments scheduled at the Large Hadron Collider of the CERN in Geneva to find more answers to this question.
Higgs Boson : The God Particle : X Boson
An hypothesis formulated in 1967 by Soviet physicist Andrei Dmitrievich Sacharov says that in the first second after the Big Bang all the forces we observe would be assembled into a superpower whose vector would be the X particle.
When the universe started to cool the X particle decayed causing the asymmetry of the matter and antimatter.
This asymmetry may have led the formation of antimatter islands, such as anti-galaxies, which would still exist at a very big distance from us.
Anti-Gravity
If we put a positive and a negative mass (matter and antimatter) in the Newton's gravity law, we will obtain a repulsive force.
So it should mean that there could be an "anti gravity" between matter and antimatter: if it's true, it is probably the cause of the asimmetry after the Big Bang, and the matter didn't really win the "fight", but it has evoluted apart from the antimatter.
It follows that there should be a part of the universe made of antimatter, but that it's very far from us because of the repulsive force between the two parts.