First systems were described as systems that had one method of storing energy.  Second order systems; wait for it.... have two methods of storing energy.  Using a similar mechanical system to the first order case, the common spring mass damper system is a good example of a second order system.

The spring and the mass are both capable of storing energy whereas the damper can only dissipate energy.

Consider the situation with the damper removed.  If a force is applied to push the mass down, then strain energy is stored in the spring.  As the mass is released, this energy is converted into kinetic energy in the mass as it starts to move.  This KE becomes potential energy as the mass gains height.  The potential energy then becomes kinetic energy which then becomes strain energy in the spring and so on.  The system oscillates as energy is transferred from the mass to the spring and back again.  In theory the system oscillates continuously.  In reality of course, the system loses its energy to the surrounding, there is resistance as the mass travels through the air and the spring generates heat and therefore the after a period of time the system does come to rest.  In many real world applications the spring mass is fitted with a damping device in order to control system behaviour and in most cases to "dampen out" oscillations.