Since Special Relativity demonstrates that space and time are variable concepts, then velocity (which is space divided by time) becomes a variable as well. If velocity changes from reference frame to reference frame, then concepts that involve velocity must also be relative. One such concept is momentum, motion energy.
Momentum, as defined by Newtonian, can not be conserved from frame to frame under Special Relativity. A new parameter had to be defined, called relativistic momentum, which is conserved, but only if the mass of the object is added to the momentum equation.
This has a big impact on classical physics because it means there is an equivalence between mass and energy, summarized by the famous Einstein equation:
The implications of this was not realized for many years. For example, the production of energy in nuclear reactions (i.e. fission and fusion) was shown to be the conversion of a small amount of atomic mass into energy. This led to the development of nuclear power and weaponry.
Likewise, the equivalence of mass and energy allowed Einstein to predict that the photon has momentum, even though its mass is zero. This allows the development of light sails and photoelectric detectors.