What's dark matter and who discovered it? I have heard about dark matter that's called the Master Of The Universe. What's this and is the dark matter the reason galaxies exist?
 A: Something else which the other answers have alluded to but not stated plainly is that, in a way, "What's dark matter?" has no real answer in the way you are inquiring. That is, no real answer other than to explain what the term means. By definition, dark matter is matter which is affecting the universe but which you cannot see. The moment you figure out what it is, it is no longer dark matter.
As an example, let's assume that the universe is full of small balls of iron or lead, densely filled with lots of them ranging in size from peas to baseballs, and we don't see any around us because each star system's activity clears them out by incorporating them into its star and planets over time the same way planets clear out other debris in their orbits. This is a silly proposal, but it is merely to demonstrate a point. These balls add enough mass to the galaxy to account for the anomalies we see, but since we cannot see the balls between stars and galaxies because we cannot detect the small objects so far away we don't know that's the cause. These balls of iron or lead are then dark matter. Fifty years from now we develop telescopes so powerful that we can see objects one inch across from light years away and we discover these balls; we just found dark matter. Now we understand it, and it's not dark matter anymore.
There are anomalies of astrophysics which have already been pointed out in the other answers. Astronomers are trying to find out what the causes are, and they are taking many different routes in doing so. There could be undiscovered types of particles. Perhaps instead we need new laws of physics.
But it's not all that fanciful. Despite my silly analogy of iron/lead balls, in reality dark matter often is just plain, old, ordinary matter that we just haven't seen yet. In years past, when we had not discovered all the planets in our star system which we know now, astronomers saw anomalies in the orbits of the furthest stars they could perceive, and these could be accounted for by other objects (planets) that they just could not see yet, which were later found. These planets were dark matter (I noticed someone else incorrectly commented elsewhere that a distinction needs to be made; but the planets were in fact dark matter) until they were found, after which they were no longer dark matter. Obviously, it would be silly to try to account for anomalies such as gravitational rotation speed by saying it's all packed away in planets we cannot see; that would be hand waving. But it could be that the reality of the situation is a combination of normal matter as we know it (but which is unseen), undiscovered types of massive objects, or possibly even new physics and a bunch of Plutos ;)
Really though, what you need to keep in mind is that "dark matter" is not really a thing that exists in the universe. It's just a term, and trying to carefully define or understand the term "dark matter" is a matter of semantics more than science. It's like when the astronomy bigwigs got together recently and decided what is and what is not a planet. Pluto got declassified not because it changed or even because our understanding of it changed, but rather because some people decided to change the definitions of some words. The "change" had no impact on actual science whatsoever; only in how we write about science in our journals and textbooks.
Likewise, it's less important to understand "what dark matter is" and just realize that it means "there's a thing there and we don't know what it is." The important part is understanding what the real anomalies are and what the hypothesis are for describing what causes them.
There's a quasi-answer for a quasi-term. I hope that helps. I'm not trying to belittle the term, rather I hope I brought you some good perspective on it.
A: It's everywhere! Astronomers have found that supermassive black holes live in the heart of galaxies and pull stars with incredible speeds, but that they are not strong enough to hold all the stars in the gigantic galaxies together. 
So what does hold them together in the spiral? Swiss astronomer Fritz Zwicky wondered why galaxies stayed together in groups, so he concluded that there must be something that no one detected before, and he named it "dark matter". 
To prove it, scientists build virtual galaxies in computers, with virtual stars, and virtual gravity. Astronomers were looking forward to this simulation experiment returning the integrated galaxy similar to our galaxy. Instead, they found it disintegrated! The gravity of the galaxy is not enough to hold it together, so scientists added virtual gravity from virtual dark matter to the simulation experiment. And it solved the problem, the gravity from dark matter held the galaxy together. 
So we can say that "dark matter is the Master Of The Universe". This is a short story about dark matter. To explain it in detail, I would need lots of books.
A: No one has discovered it. Dark matter is a proposed explanation to some observed phenomena.
In particular, 


*

*Galaxies rotate at a speed that implies they are quite heavy, especially towards the outer edges - but when we look at the mass from stars and interstellar gas, there isn't enough to make them spin the way they do.

*Gravitational lensing is a phenomenon in which light is bent by the presence of lots of mass. Usually, we see the mass doing the bending - a big star cluster or something. But some gravitational lenses don't have enough visible matter associated with them - there must be some missing mass involved.

*Analysis of the light produced from the early-universe - a few hundred thousand years after the big bang - suggests strongly that there is some sort of mass clumping in a complicated way, and that these clumps don't interact with light strongly (or at all)
All of these point to some missing, heavy matter which is, apparently, invisible.
We call this hypothesized invisible substance "dark matter," because it is literally dark (we can't see it) and has mass (so it's matter).
Lots and lots of people are looking for dark matter in all sorts of ways, and many are also seeking to explain it theoretically. None have succeeded so far, but the search is continuing.
You ask if this is "the reason galaxies exist." This is not how I would phrase it, but in a way, yes. A real problem in cosmology is "why isn't the universe perfectly uniform following the big bang?" and for complicated reasons, this can be solved with dark matter. 
