The capacitor is initially charged. When the circuit is closed, the capacitor voltage is applied to across the inductor. The nature of a inductor means that the current increases proportional to the applied voltage. With the capacitor voltage applied across the inductor, the inductor current starts going up.
As the inductor current goes up, that current is draining the capacitor, which reduces the voltage across the inductor. This means the current goes up more slowly over time. When the capacitor voltage reaches 0, the current stops going up altogether.
Now there is current but no voltage. That current continues to decrease the capacitor voltage, now making it negative. That negative voltage across the inductor now reduces the current. The current reduces more quickly as the negative voltage builds up on the capacitor. Eventually the current is reduced to 0. At that point, the voltage on the capacitor is the negative of what it started at.
Now we have the same conditions as we started with, except that the capacitor voltage is negative instead of positive. The same thing will happen as happened up until now with all the signs flipped. The result of this is the current is again 0, but the capacitor voltage has flipped again, this time resulting in what it started with when the switch was first closed.
With ideal components, this process repeats indefinitely. The finite energy is constantly sloshed back and forth between being voltage on the capacitor and current thru the inductor. Both the voltage and current have sinusoidal profiles. This is referred to as resonance.
This is also called a tank circuit. Tank circuits are often used as notch filters, since the frequency of oscillation is specific to the particular capacitance and inductance values. It has the interesting property that the impedance across the parallel combination of capacitor and inductor is 0 at 0 and infinite frequency, and infinite right at the resonant frequency. This drastic impedance peak at the resonant frequency is often exploited to either filter in or filter out that frequency.