my understanding is it refers to the frequency at which an object will naturally oscillate when provoked.
eg. hit the wine glass and the tone you hear will be its natural frequency.
better example is a tuning fork. bash it against anything and it will resonate at its natural frequency allowing you to tune your instrument.
additionally, provide energy to the object at its natural frequency and the oscillations within the object will continue to grow, similar to pushing a child on a swing at the same frequency at which it is swinging.
eg. play a sound next to a wine glass of the same frequency as its natural frequency and the oscillations will continue to increase to the point where the wine glass shatters. hence the stories of operah singers shattering wine glasses with their voices.
For an object to oscillate there must be what's called a restoring force acting on it. A restoring force pushes or pulls the object back toward an equilibrium point (usually where it was before it was "provoked" into oscillating) at which the force acting on it goes away. But now the object is moving, and objects in motion keep moving when there's no force on them. So the object goes past the equilibrium point in the other direction, which in turn causes the restoring force again, this time pushing/pulling in the other direction back toward the equilibrium point. And so on....
In most cases, the natural frequency depends on the mass of the object (objects with greater mass usually oscillate at lower frequencies) and the strength of the restoring force (stiffer objects oscillate at higher frequencies).
If you do a google on "modal analysis" you will get more than you hoped for.
Just to add a bit, the natural frequency is only one of many that you will induce if you plink a glass or provide an impulse to induce the vibration. All of the orders of that natural frequency will also be excited at one time assuming enough energy is imparted. Through the use of a spectrum analyzer and instrumentation they can be separated.