So, what is antimatter?
Even from the name it is obviously the "opposite" of ordinary matter, but what does that really mean?
As it happens there are several equally valid ways to describe the difference. However, the one that I think is easiest to explain is that in antimatter, all of the electrical charges on all of the particles, at every level, have been switched around.
Thus ordinary electrons have negative charges, so their antimatter equivalents have positive charges. Protons are positive, so in antimatter they get the negative charges. Even neutrons, which have no overall charge, still have internal parts (quarks) that very definitely have charges, and those also get flipped around.
Now to me the most remarkable characteristic of antimatter is not how it is differs from ordinary matter, but how amazingly similar it is to ordinary matter. It is like an almost perfect mirror image of matter -- and I don't use that expression lightly, since it turns out that forcing ordinary matter into becoming its own mirror image is one of those other routes I mentioned for explaining what antimatter is!
The similarity is so close that large quantities antimatter would, for example, possess the same chemistry as ordinary matter. For that matter there is no reason why an entire living person could not be composed of antimatter. But if you do happen to meet such a person, such as while floating outside a space ship above earth, I strongly recommend that you be highly antisocial. Don't shake hands or invite them over, whatever you do!
The reason has to do with those charges, along with some related factors.
Everyone knows that opposite charges attract. Thus in ordinary matter, electrons seek out the close company of protons. They like to hang out there, forming hydrogen. However, in ordinary matter it also turns out that there are also all sorts of barriers -- I like to think of them as unpaid debts to a very strict bank -- that keep the negative charges of electrons from getting too close to the positive charges of the protons.
Thus while the oppositely charged electrons and protons could in principle merge together and form some new entity without any charge, what really happens is a lot more complicated. Except for their opposite charges, electrons don't have the right "debts" to pay off everything the protons "owe," and vice-versa. It's like mixing positive apples with negative oranges. The debts, which are really called conservation laws, make it possible for the powerfully attracted protons and electrons to get very close, but never close enough to fully cancel out each other's charges. That's a really good thing, too. Without that close-but-not-quite-there mixing of apples and oranges, all the fantastic complexity and specificity of atoms and chemistry and biochemistry and DNA and proteins and us would not be here!
Now let's look at antimatter again. The electrons in antimatter are positively charged -- in fact, they were renamed "positrons" a long time ago -- so like protons, they too are strongly attracted to the electrons found in ordinary matter.
However, when you add electrons to positrons, you are now mixing positive apples with negative apples. That very similarity turns out to result in a very dangerous mix, one not at all like mixing electrons and protons. That's because for electrons and positrons the various debts they contain match up exactly, and are also exactly opposite. This means they can cancel each other's debts all the way down to their simplest and most absolute shared quantity, which is pure energy. That energy is given off in the form of a very dangerous and high-intensity version of light called gamma rays.
So why do electrons and positrons behave so very badly when they get together?
Here's a simple analogy: Hold a rubber band tightly at its two ends. Next, place an AAA between the strands in the middle. (This is easier for people with three arms.) Next, use the battery to wind up the rubber band until it is quite tight.
Now look at the result carefully. Notice in particular that the left and right sides are twisted in opposite directions, and in fact are roughly mirror images of each other.
These two oppositely twisted sides of the rubber band provides a simple analog to an electron and a positron, in the sense that both store energy and both have a sort of defining "twistiness" that is associated with that energy. You could easily take the analogy a bit farther by bracing each half somehow and snipping the rubber band in the middle. With that more elaborate analogy the two "particles" could potentially wander off on their own.
For now, however, just release the battery and watch what happens. (Important: Wear eye goggles if you really do try this!) Since your two mirror-image "particles" on either side of battery have exactly opposite twists, they unravel each other very quickly, with a release of energy that may send the battery flying off somewhere. The twistiness that defined both of the "particles" is at the same time completely destroyed, leaving only a bland and twist-free rubber band.
It is of course a huge simplification, but if you think of electrons and positrons as similar to the two sides of a twisted rubber band, you end up with a surprisingly good feel for why matter and antimatter are dangerous when placed close together. Like the sides of the rubber band, both electrons and positrons store energy, are mirror images of each other, and "unravel" each other if allowed to touch, releasing their stored energy. If you could mix large quantities of both, the result would be an unraveling whose accompanying release of energy would be truly amazing (and very likely fatal!) to behold.
Now, given all of that, how "real" is antimatter?
Very, very real. Its signatures are everywhere! This is especially true for the positron (antimatter electron), which is the easiest form of antimatter to create.
For example, have you ever heard of a medical procedures called a PET scan? PET stands for Positron Emission Tomography... and yes, that really does mean that doctors use extremely tiny amounts of antimatter to annihilate bits of someone's body. The antimatter in that case is generated by certain radioactive processes, and the bursts of radiation (those gamma rays) released by axing a few electrons help see the doctors see what is going on inside someone's body.
Signatures of positrons are also remarkably common in astrophysics, where for example some black holes are unusually good at producing them. No one really understands why certain regions produce so many positrons, unless someone has has some good insights recently.
Positrons were the first form of antimatter predicted, by a very sharp fellow named Paul Dirac. Not too long after that prediction, they were also the first form of antimatter detected. Heavier antimatter particles such as antiprotons are much harder to make than positrons, but they too have been created and studied in huge numbers using particle colliders.
Despite all of that, there is also a great mystery regarding antimatter. The mystery is this: Where did the rest of the antimatter go?
Recall those debts I mentioned? Well, when creating universes physicists, like other notable entities, like to start the whole shebang off with pure energy -- that is to say, with light. But since matter has all those unbalanced debts, the only way you can move smoothly back and forth between light and matter is by having an equal quantity of antimatter somewhere in the universe. An amount of antimatter that large flat-out does not seem to exist, anywhere. Astrophysicists have by now mapped out the universe well enough to leave no easy hiding places for large quantities of antimatter.
Recall how I said antimatter is very much like a mirror image of matter? That's an example of a symmetry. A symmetry in physics is just a way of "turning" or "reflecting" or "moving" something in a way that leaves you with something that looks just like the original. Flipping a cube between its various sides is a good example of a "cubic symmetry," for example (there are fancier words for it, but they mean the same thing). Symmetries are a very big deal in modern physics, and are absolutely critical to many of our deepest understandings of how our universe works.
So matter and antimatter form an almost exact symmetry. However, that symmetry is broken rather spectacularly in astrophysics, and also much more subtly in certain physics experiments. Exactly how this symmetry can be broken so badly at the universe level while being only very subtly broken at the particle level really is quite a bit of a mystery.
So, there you have it, a mini-tutorial on both what antimatter is and where it occurs. While it's a bit of overkill, your question is a good one on a fascinating topic.
And if you have read through all of this, and have found any of what I just said interesting, don't just stop here! Physics is one of those topics that gets more fascinating as you dig deeper you get into it. For example, some of those cryptic-looking equations you will see in many of the answers here are also arguably some of the most beautiful objects ever uncovered in human history. Learning to read them well enough to appreciate their beauty is like learning to read great poetry in another language, or how to "hear" the deep structure of a really good piece of classical music. For physics, the reward is a deep revelation of structure, beauty, and insight that few other disciplines can offer.
Don't stop here!