Quick answer, I hope I get it right:

Everyday matter is made of a variety of "fundamental" particles - you'll have heard of electrons, protons, and neutrons, and there are many others which are -not- everyday.

(For instance, an atom is a nucleus of one or more protons, an approximately similar number of neutrons - except for a hydrogen atom which can have one proton and between 0 and 2 neutrons - and one electron for each proton. Technically, if the number of electrons is different - which can happen in an ordinary chemical reaction - it's not an atom but an ion. For instance, table salt, sodium chloride, consists of sodium ions and chlorine ions.)

The "fundamental" particles mostly seem in fact to be made up of particles called "quarks", but the last that I heard, no one had found a single quark. (But that was some time ago, and they wouldn't necessarily come tell me.) So we can skip that.

Anyway, the everyday particles have twin particle types that are much more rarely seen - an antiproton, an anti-electron (positron), an antineutron. Those particles behave like the more familiar particles but have opposite electric charge to their twin (electrons and protons have electric charge - that's what keeps electrons on atoms, opposite charges attract), some other opposite properties, and if a particle meets its own anti-twin particle type then the particles cease to exist, producing energy - according to Einstein, that energy was in there all along, that's the famous E = m c squared.

By the way, anti-particles don't travel backwards in time or anything funny like that, except in bad science ficttion stories. They are just like ordinary matter, but don't try to pick them up, you'll be sorry. (Actually, any significant quantity of anti-matter would explode in air.)

Anti-particles are produced in some nuclear reactions - don't ask me to explain why, but some radioactive elements release anti-particles, and I think they appear in the "solar wind". I think it makes sense in quarks.

Scientists have been puzzled why the universe (as far as we see it) has only the everyday particles, since their opposites seem to be just as good. A planet could be made of anti-matter if there was enough anti-matter. But if there were both sorts of common particles around, our kind and the anti, around, then we'd see spectacular explosions where they meet. As it is, even anti-atoms don't exist in nature.

I think the current theory for this has some positive evidence - that in fact the anti-particles of everyday matter particles are -not- just the same as their twins, and there is a small bias against anti-particles. So after a whole lot of simultaneous creations of particles and corresponding anti-particles from energy (I forgot to mention that, didn't I?), at the beginning of the universe, there are lots of particles left over in the universe without an anti-particle made at the same time. And those are the particles that Earth and human beings are made of. At least, those protons, neutrons, electrons. Mostly.

However, the last I heard about -that-, they couldn't get the calculation to come out right and use the theory to "predict" the universe as it actually exists.