This might be an obvious question, but I was wondering how a Tesla coil emits sparks that's seem to complete a circuit in thin air, and how one end of the secondary completes a circuit with the other side of the secondary. Is this circuit simply being completed by the electromagnetic radiation in the air reaching the ground and completing this circuit or is there more to it?
The circuit is completed by capacitance to ground.
Ground in this sense is all that is conductive in the vicinity of the topload. When you consider that you have 100s of kV available, almost anything is conductive. This includes wiring in the walls and ceiling, central heating pipes, building materials with any degree of moisture in, aerial leads, data and telephone leads, your garage door, any steel frame to the building. It also includes ground, if you're running in a ground floor apartment, or out of doors.
The base of the secondary is connected to ground. By choosing what you connect it to, and how, you choose (to some extent) where the return currents will flow. This controls now much EMI any discharges and strikes will inject into your house wiring.