In section 4-4 of Six Not-So-Easy Pieces, Richard Feynman describes the inelastic collision of two masses moving at relativistic speeds (a significant fraction of the speed of light).  The two masses, moving in opposite directions, meet and stick together.  Because they are moving so fast, their masses have increased beyond their masses at rest, as required by special relativity.  When they collide and stick, the resulting mass isn't moving (because the incident masses had equal and opposite momentum), but its total mass is the sum of the two enhanced masses.  That is, greater than the sum of the two original rest masses.  Feynman doesn't say so explicitly, but this appears to be conversion of the original particles' kinetic energy to mass.  Whether such an event can actually be caused, I have no idea.

What is 'matter?'

Therein lies the answer.