First, imagine a swing set: a swing is essentially a pendulum. Every pendulum has a natural frequency defined by the length of its rope or chain. When you push on a child in a swing with a frequency close to the natural frequency, the child keeps getting pushed higher and higher -- until your arms can't push any harder. If you try pushing at any other frequency, you'll find that the child won't go very high (and you'll probably get yelled at).
Essentially, a capacitor stores energy in the form of an electric field. An inductor stores energy in the form of a magnetic field. When you connect the two together, energy can slosh back and forth between them, much like a pendulum swinging. The amount of capacitance and inductance determines how fast this sloshing takes place -- just like how the length of a pendulum's rope determines how fast it naturally wants to oscillate. By themselves, a capacitor and inductor do not produce any energy -- they only serve to produce a resonance so that frequencies other than their natural resonant frequency are attenuated. This is the basis of any oscillator: make one frequency "preferred" while others are attenuated. If the system is very sensitive, it will begin oscillating on its own, reinforcing itself.
- Warren