If you took even one high school science class, you recognize matter as the stuff things are made from. If you were an 'A student', you remember it as "that which has mass and takes up space". In short, matter is stuff you can touch - rocks, cars, people, buildings - it's all composed of matter.
The thing is, that definition we all learned is so simplified that it's almost untrue. Yes, we can touch matter, but there are also many more types of matter. And bringing mass and volume into the equation just confuses the issue.
To be honest, there's no universally-accepted definition of matter. You were probably taught that mass is the amount of matter and volume is the amount of space it takes up, but that's like saying mass and matter are the same thing - they're not.
Think of a cubic foot of bricks and a cubic foot of feathers - they both have the same volume, but very different masses - and at a subatomic level, they are both composed of the same kind of matter. Which one would you rather be hit in the head with? And then there's energy - which has mass - is it matter? Does it matter?
Early definitions of matter included everything that is made of molecules - so then what are atoms? So we expanded it to include everything made of subatomic particles - so then what are they?
Today, ordinary matter is thought to be anything composed of up and down quarks, electrons and neutrinos - but then certain types of bosons are not made of quarks, even though they have mass. So do we have mass without matter?
Well, we have baryonic matter, which contains three quarks; we have degenerate matter, which is the state of fermion gas at near - ok, where pressure depends upon number of fermions rather than temperature; and we have strange matter, which is basically a quark liquid that includes strange quarks alongside the usual up and down. Got all that?
And that's not all! Now we get weird (science is great!). Remember Bizarro Superman? Well, subatomic particles have their bizarro brethren too - they're called antiparticles. And they make up antimatter. Antimatter is not the opposite of matter - that would be the absence of matter - antimatter is the anti to ordinary matter.
If matter and antimatter meet, they destroy each other instantly, becoming other particles of equal collective energy. Antimatter isn't found on Earth because there's matter everywhere - it would blow up instantly, and the explosions are pretty large and dramatic. That's why movies with antimatter bombs are both unrealistic and terrifying at the same time.
Then there's dark matter, which we can't see or measure, but by golly, we know it's there! How? Because it affects the things around it via gravitational pull, which we can measure. But the dark matter itself doesn't really emit anything we can observe directly, so we have to infer its presence by measuring everything around it.
The best we have so far is that it's composed of particles that only form in extreme high energy, i.e: the Big Bang, and still exist from that single event. Remember thermodynamics? Matter can't be created or destroyed. The rest we're leaving to the astrophysicists.
In case things aren't weird enough for you yet, there's also exotic matter - or at least, the particle physicists think so. It's really just a catch-all term for things that don't fit into any categories. Things that we only know of because the math says they must exist. Things that have negative mass and are repelled by gravity.
Yes. High school science may have been boring, but you were only scratching the surface. Once you get down and dirty, even simple things like defining matter will totally blow your mind.