A : when you want to power a small LED without ability of controlling it (for example power on led). The current going through the LED will be (Vpsu - Vled)/R
B : you're relying on the transistor to limit the current. You can use the hFe of the transistor and pick a resistor to "open up" the transistor just enough current flow through, therefore limiting the led as well.
hFe varies with transistor, it's usually within a wide range, and it also varies with temperature, so it's hard to pick one resistor that will work with any transistor out of - let's say - a batch of 100 transistors. You also have to be careful about how much power is wasted inside the transistor, and keep in mind that as the transistor heats up, the hFe will also change, so the current limit as a consequence will change.
It's worse than A because you're also wasting power in the resistor connected to the base (well, base-emitter), but it's a cheap solution and useful to turn on and off stuff for short periods of time.
For example, a microcontroller can output in general 20-25 mA though any pin, so if you have a LCD backlight (without its own current limiting system) that's supposed to use 50mA, you can use this method to power the LCD backlight from power supply, and the mcu pin will only use 0.1-0.5mA, enough to partially open up the transistor.
C: the transistor is opened fully, it's not limiting the current. The resistor is there to just set a high limit for the current going through transistor from base to emitter (without resistor on base, you kill the transistor). There is some voltage drop on the transistor from Collector to Emitter but it's generally small (0.2-0.7v) so your new led current limit is (Vpsu - Vled - Vnpn)/R
An alternative to this C method would be to use a mosfet, which will waste less energy as there's no current from base to emitter as is the case with transistors. You would still need a resistor from gate to ground to turn off the mosfet when you want to kill the connection but it can be a large resistor (10 kohm or more). There's some capacitance otherwise which will keep the mosfet working even when your microcontroller stops sending voltage to gate.
There's also some resistance between Drain and Source, but it's usually small enough to not matter in formulas (a cheap BS270 for example has 1.5-2 ohm resistance) so you still have to use a current limit resistor with the led, and keep in mind the mosfets power dissipation.