Most people have played with a slinky before, it goes down as one of the most classic yet simple toys of all time probably. My dad told me the other day about it being the 70th anniversary of the slinky being up for public sale. The story goes, the inventor - Richard James - thought of the idea when he was using springs to create instruments to stabilize boats in rough seas. While doing this he accidentally knocked a spring off of a shelf and watched as it fell down the stairs in a graceful manner as opposed to tumbling down.
The Slinky demonstrates the effects of friction and inertia, potential and kinetic energy. Since inertia determines how resistant an object is to a change in motion, this clearly has pertinence in the motion of a slinky. This resistance to a change in motion, which is greater in metal slinkies than plastic ones, keeps the object moving down the stairs. Friction plays a role in the motion of the slinky as well because as the slinky falls down the stairs, the bottom of it does not move when it hits the next step, thus containing the object's momentum on the top part of the slinky - propelling it to the next step. There's also a clear transfer between potential and kinetic energy in the slinky's fall. As the slinky starts with an impulse from its rightful owner, it has potential energy in relation to the next step down. Once the slinky makes contact with the next step this is converted to kinetic energy which will propel it to the next step, and so on.
All in all, the physics behind the slinky is relatively simple, but no one can deny that it's fun to push one down the stairs and watch it go.