Take a slinky and stretch it out so that it's fairly taut - across a room, say. Wiggle one of the ends up and down. Notice how whenever you disturb part of it, the disturbance moves away down the length of the slinky?
This is what in physics we call a wave - a disturbance that moves. There's a whole bunch of interesting stuff that happens with it that you can play with (what happens when the disturbance reaches the end of the slinky? is a good place to start), but there's one thing in particular I want to draw your attention to.
Disturb the slinky again and pay close attention to the little loops that compose the spring. What you should notice is that while the disturbance as a whole travels away, the individual pieces of matter basically only move back and forth in line with however you disturbed it; up and down, side to side, whatever. This shouldn't be a huge surprise, right? It's not like you grabbed part of the slinky and threw it across the room or something. The particular loop of the slinky you disturbed just pulls on the neighboring loops, which pull on their neighbors, and so on; it's a function of the fact that the slinky is springy and tries to return to its original shape due to tension forces. The "wave" isn't an actual object, it's just a description of the process as a whole.
So then what happens if you set up something, like a domino chain or something, next to the far end of the slinky and then disturb the slinky sideways? You can try it if you like, though what happens is essentially what you'd expect - the wave propagates down the slinky and runs into the dominoes and knocks them over.
But we just said that the wave isn't a physical object. It's just a little disturbance in the force (heh heh); you just accomplished the same thing as if you'd thrown a baseball down to the other end of the room to knock the dominoes over, only the only thing that moved across the room is a mathematical description of the force and energy transference taking place in the slinky. No actual, physical object crossed the room at any point.
Of course, you could argue that the same thing happens if you just stretched out a chain of dominoes across the room to reach the domino on the other side, but each individual falling domino moves slightly towards the next one. There's a net motion involved in the right direction. But with the slinky, each little bit of matter only moves from side to side (if you were careful). Not only does no object cross the room, no object even moves in the right direction to cross the room.
Congratulations. You just affected an object a whole room away from you with nothing but the power of math.