A natural frequency is a frequency at which an "elastic" object will naturally oscillate, without the introduction of any outside force, or at least that's how it's defined in physics. A way of demonstrating the idea is with a pendulum. If a pendulum is put into a swinging motion and left alone, it will track according to a certain number of swings per minute, which is the natural frequency of the pendulum. Simple objects and complex systems can have a natural frequency. A wine glass, for example, would also reveal its natural frequency if tapped gently to produce a ringing sound.
Resonance has to do with the interaction of frequencies. A demonstration of this phenomenon is an experiment done with a wine glass and an opera singer. This effect, known as an example of acoustic resonance, involves the singer hitting a note, the pitch of which matches the natural frequency of the wine glass. Since objects are apt to absorb vibrations that match their natural frequency, if the vibrations from the opera singer's note (which are in the air) match that of the glass (which is touching the air), then the glass will absorb those vibrations until the glass cannot absorb any more, and it shatters. The shattering is due to the structural integrity of the glass being unable to store more kinetic energy, like a battery being overcharged and exploding.
Does any of this remind you of your relationships with others? Take this quiz, and we'll identify your natural frequency!