Hmm, it's a tough question. Of course, the first thing that comes to my mind is Bill Nye: "inertia is a property of matter" and so I would say that matter is anything that would require force to change it's momentum. But that's not really a good explanation, in my mind.
I can't think of any better ones, but I'm very curious to see what answers you get. Quick question: are you talking about matter vs energy? Matter vs waves? Matter vs antimatter? That might help narrow down answers.
This reminds me of another question: what is a number? We use them all the time, and seem to have a good understanding of how they work, but can we really say what they are precisely? I've heard different explanations, but none have truly satisfied me.