Before I begin, I want everyone to know that I take safety very seriously. Tesla coils are, with no doubt, extremely dangerous. The Pupman Safety Site has been useful.
"Consider what happens when you don’t drive the circuit (assume that the AC source in the figure is replaced by a wire), but start out with the capacitor charged. The capacitor wants to discharge, so charge flows around the circuit, through the inductor, to the other plate. In the process, a magnetic field builds up inside the inductor. When the charge on each plate of the capacitor is zero, current stops flowing. But at this point, the inductor has energy stored up in a magnetic field - which tends to oppose change. The magnetic field collapses, inducing a continuing current in the same direction, thereby recharging the capacitor and restarting the cycle in the opposite direction."