Lithium ion batteries recharge in two phases: Constant Current (CC) followed by Constant Voltage (CV). If you are familiar with electronics, you will know that a typical bench power supply will be able to operate in constant current and constant voltage modes, so think of the Supercharger as a giant power supply! (They can also operate at constant power, but that is not used when charging a battery).
In the CC phase the car will call for a certain amount of current, and the Supercharger will supply what it can (it may be less than that requested if for example, someone is using the adjacent stall and you are sharing the same power supply--more on this in a moment). The voltage will float to whatever voltage the battery is at. Generally this will slowly rise as the state of charge increases, so the power (current * voltage) will gradually rise as well, but not significantly. In rough figures though, the pack voltage will be around 400V, so if you are achieving 120kW charging, the Supercharger is putting out 300A. Like I said though, while the current might be constant at 300A, the voltage and power will slowly rise.
Once the pack voltage reaches a certain threshold (determined by the battery management system software in the car), the car will call for a switch from CC to CV mode and call for a specific voltage (which will be slightly above the pack voltage at that point). The Supercharger will then supply whatever current naturally flows into the battery. As the battery becomes fuller, that current will taper off (significantly at the end), and the resultant power being transferred will likewise drop. As ggr pointed out above, this is the tapering effect. Generally this is why fast charging speeds are typically quoted to 80%, because above that you go into CV mode and the charge rate drops pretty quickly. To use the water analogy, imagine you had a water source at a given height (voltage) with a rigid hose attached to an outlet spigot. The other end of the hose sits floating in a second bucket of water (the battery) several feet away. At first the water level of battery bucket is low and the slope from the water source to the battery is pretty steep and the water runs fairly fast. But as the water level of the battery rises and the slope becomes less steep, the water flows slower and slower until it's just barely dripping out the other end.
Once the current drops below a certain threshold (again, determined by the battery management software in the car), it is considered "full" and charging terminates.
Recall I mentioned sharing adjacent stalls. The Supercharger is architected such that adjacent stalls with the same number (e.g. 4A and 4B) share a single power supply. This greatly reduces the cost per stall of a Supercharger site, and is done on the assumption that as one car is finishing off and in CV tapering mode, the other car can at least receive most of the requested current in CC mode. For example, if the SC is capable of supplying a total of 150kW between the two stalls and one car is finishing up and only drawing 45kW, that leaves 105kW for a new car that pulls up and plugs into the adjacent stall. So it won't quite get the 300A it's requesting, but it might get 262A. So you should avoid plugging into an adjacent still if you can help it, but if you have to, you should certainly use the one which the connected car is farthest along in its charging (if you can determine that).
Oh, and since you already own a non-Tesla electric car, if it has quick charging capability, it basically works the same.
