What is a wave?

A wave is the effect of energy propagating in a medium that has some elasticity through sinusoidal vibrations of the medium. These vibrations can be either perpendicular to the direction of energy propagation or also longitudinal. Sometimes the energy is "trapped" in space in a pattern,one then has a "standing wave".

Waves in a lake have both components: the medium vibrates and displaces incrementally the water, the molecules staying in place. Sound through a solid, the solid vibrates in place and the propagated energy activates the air according to the frequency it vibrates on. A chord on a violin has a standing wave pattern on the frequency it is tuned on, which generates sound waves, vibrations of air.

These were the waves known until special relativity arrived on the scene.

It was thought by the wise men of the time that since electromagnetic waves were waves par excellence, displaying interference etc, then there must be a medium on which they propagate, and they called it ether: a universal rest frame, everything moving moved through ether. The Michelson_Morley experiment (see link above) which measured the constant speed of light showed experimentally that there is no ether, no universal frame of reference.

So electromagnetic are the waves that need no medium to propagate the energy on. They are self sustaining: the electric and magnetic fields changing in time create the form of the wave and energy propagates at the speed of light, perpendicular to the direction of sinusoidal change of the electric and magnetic fields.

These last are the waves on which gravity waves are mathematically patterned on: changes on the gravitational field (space curvature in this case) which do not depend on an absolute reference frame but propagate the energy at the speed of light by pushing ahead the sinusoidal gravity field changes .

A wave is anything that moves like a wave!

Wave is useful in physics because so many continually varying things can be modeled as a wave (generally a sin wave or set of sins)

Often it isn't anything that is moving. In a speaker a pressure wave - basically a region where the air molecules are closer together - is what moves from the speaker to you. No actual molecules move very far. This is the same as water waves, although the surface moves up and down - the water doesn't move forward as a mass (this is obvious if you watch ducks floating on the surface)