Well, a good notion of what antimatter is involves understanding what matter is, so let's start with that. Our best current understanding of matter, at the most fundamental level, is of matter as excitations of a field. What does this mean? Well, for an introductory example, think about, say, air: if we take a blob of air, at every point in it we can talk about the pressure of the air at that point. That means that we can define something called the pressure field of the air blob - the pressure field is the pressure of every point in the blob, at every moment in time. Now, if someone disturbs the air on one side of the blob, a wave of pressure is created, and this pressure wave travels from one side of the blob to the other (this is what sound is). The pressure wave is what we call an "excitation" of the pressure field.

This idea is what we mean when we say that "matter consists of excitations of a field." The universe contains certain fields (like the pressure field, but not "made up" of anything else like air), and when those fields are excited (that is, when we put some energy into those fields), we get excitations that travel through the field; those excitations are what we call particles. For example, you may have heard of the electromagnetic field; photons (particles of light) are excitations of the electromagnetic field. Of the various fields we know (or think) exist; we call some of them "matter" fields. So, that's what matter is: excitations of a fundamental field.

So now, what's antimatter? Well, it's exactly the same thing: antimatter is also the excitation of one of these fields, no different from ordinary matter. The reason for calling it "anti" matter is that it's a different sort of excitation of the field that can interact with "normal matter" excitations so that the excitations annihilate and create energy (usually in the form of photons). If we call the excitations "particles," then we would say that a particle and an antiparticle annihilate to create photons.

One important fact is that as far as we currently know, there is absolutely no reason that matter should be preferred over antimatter; it's just as possible that the entire universe could have been made of antimatter instead of normal matter (in which case we would have referring to antimatter as normal matter, and normal matter as antimatter!). The reason that the universe is made up almost entirely only of matter is a mystery that physicists are actively trying to shed some light on.

The easiest definition of antimatter is that it is just like matter, except it is made up of particles with the opposite charges. This means that instead of protons, which are particles with a mass of 1.007 atomic mass units (amu) and a positive charge, antimatter has nearly identical particles called antiprotons with the same mass and a negative charge. Instead of electrons with a negative charge and a mass of 0.00055 amu, antimatter has positrons which have a positive charge but the same mass.

However, protons, neutrons, and electrons are not the smallest units of matter that we know about. Instead, they are all made up of even smaller particles called quarks. Quarks are elementary particles; this means that they are the smallest particles known to exist. All matter is made up of quarks and all antimatter is made up of antiquarks. Quarks and antiquarks have the same mass, lifetime, and spin, but different electric and color charges. (Although you have probably never heard of spin and color charge, they are important physical properties of particles just like mass and electric charge.) Larger particles are made of quarks and larger antiparticles are made of antiquarks. There are many types of particles in the universe and each one has a corresponding antiparticle.

Antimatter and matter annihilate when they meet. For example and electron and a positron would destroy each other and release large amounts of energy in the form of gamma rays. In fact, positron annihilation is used in medicine in the form of Positron Emission Tomography (PET) to scan images of the human body.

For more information on antimatter, here is an interesting article written for a general audience:

If you are curious about different types of particles, here is a chart will all of the particles in the standard model:

Antimatter is an exotic type of matter where everything is backwards. In normal matter, atoms are composed of subatomic particles: protons and electrons (and neutrons). Protons have a positive charge, and electrons have a negative charge. These combine together to make atoms. In antimatter, the charges are opposite (the spin of these particles is also flipped, but don't worry about that right now). So an antiproton has a negative charge, and an antielectron (called a position) has a positive charge. These antiparticles combine in a similar fashion to make anti-atoms. For example, the neutral hydrogen atom is made of one proton (+) and one electron (-). On the other hand, a neutral anti- hydrogen atom is made of one antiproton (-) and one positron (+).

To be definite let's think about the electron and it's antiparticle the positron. The electron and positron are two particles that have the same mass and opposite charge. What's interesting about these particles is that when they collide into each other both particles disappear and are replaced by a burst of gamma rays (very high frequency light). This process can also happen in reverse, sometimes a burst of gamma rays will turn into an electron and a positron. The way these facts were discovered was first by watching cosmic rays (gamma rays from outer space) and looking for positively charged particles with a mass equal to the electron mass. Positrons are also produced in some nuclear reactions and they have also been made in particle accelerators. In 2010 scientists at CERN made an "anti-Hydrogen atom," which is an Hydrogen atom made out of antimatter (i.e. one positron and one anti proton bound by electric attraction).

In many ways matter and antimatter are like identical twins and no one completely understands why our universe contains so much matter and so little antimatter. Why shouldn't there be equal parts matter and anti-matter everywhere? Or parts of the universe made of matter and other parts made of antimatter (we think this is not the case because if it were we would see gamma rays coming from the boundary between the two regions)? No one knows the complete answer to these questions yet.

So those are the experimental facts, now let me tell you a little about the theory of antimatter.

The theory of antimatter started with Paul Dirac, one of the greatest physicists of the 20th century. In 1928 Dirac was trying to come up with a theory of the electron that incorporated quantum mechanics and special relativity. Quantum mechanics and relativity are the two pillars on which modern physics is built and it's been a long standing challenge to unify them into a single theory. Today we're still trying to figure out how to unify quantum mechanics and general relativity, but in 1928 Dirac was only considering special relativity.

Dirac succeeded in writing down a theory of electrons that included both special relativity and quantum mechanics, but his theory also included these strange particles that seemed to have negative mass. There are big problems with having negative mass particles so Dirac at first guessed that these particles didn't really exist. He thought they were just some extraneous bits of mathematics hanging onto his electron theory. But then he and others quickly realized that these particles were an essential part of theory and couldn't just be thrown away. This looked pretty bad, but Dirac's theory also correctly explained some things about electrons that no one had every been able to explain before. So, Dirac stuck with it and came up a very creative idea that became known as the "Dirac sea" to explain why no one had every seen a positron (because in 1928 no one had!).

Today Dirac's original interpretation of his theory has fallen out of fashion (and we believe that antimatter doesn't really have a negative mass in any meaningful way), nonetheless the Dirac equation is still an important part of Quantum Field Theory and his results have taught us a lot about unifying quantum mechanics and special relativity, including that we almost always seem to get antimatter.