LEDs are a type of diode, and diodes experience non linear current increase with a voltage increase. If you apply a voltage in the wrong direction, they act as if they have incredibly high resistance, and block current. If you apply it in the forward direction, as you increase the voltage, not much current will flow until you reach a "knee" point in the current vs voltage curve, at which point the diode will start to behave as if it had very low resistance(allow much more current to flow and the LED to turn on). Until you reach the minimum forward voltage of the diode though, little or nothing happens. When you put the LEDs in series, you force the same current to flow through all of them, which causes the voltage to be roughly evenly split between them(as they are identical LEDs). When you added the fourth LED, you caused the voltage each LED receives to be lower than its minimum forward voltage, thus no current flows. I'm assuming you also got all 4 LEDs in the right direction as adding one backwards will also cause them not to light.
Note that LEDs have a negative temperature coefficient, which means as they heat up, they allow more current to flow, which heats them up, causing more current to flow, and this cycle, if not interrupted, will continue until the LED burns out. For this reason LEDs are installed using either a resistor in series to the LED(wastes power, but the positive temperature coefficient of the resistor compensates for that of the LED) or a current controlling circuit. Sometimes in handheld battery powered devices the resistance and PTC of the battery are used in lieu of a physical resistor. This is also the reason you can expect LEDs in parallel to differ a bit in brightness. If they're matched LEDs, they share current decently as long as none of them heat up. If one heats more than the others, it will draw more current than the others and heat up more yet.
It's better to think of LEDs in terms of current than voltage. Voltage is the pressure, or how hard the electrons are "pushing" and current is the rate of flow of electrons, or the quantity passing through something per unit of time. Current determines brightness of the LED, how hot it gets and how quickly it burns out. Voltage determines current, but only when you take effective resistance into account. Because the effective resistance changes with temperature, voltage is not the best term when looking at questions of brightness or time before failure.
As for why one, two or three in parallel would seem similar brightness, likely the difference was significant, just not visible to your eye. You mention "A resistor" and it's not clear if you used the resistor when performing this experiment, but three 3 point something volt LEDs in series connected to 9v will apply roughly 3V to each, which will likely keep the LEDs happy. In the long term, adding a resistor or a current controller is necessary to prevent the LEDs from overheating, running over current and burning out. 2 LEDs at 9V puts about 4.5V across each of them, which will cause them to run overcurrent and heat up(which will cause higher brightness, but not much because LEDs are less efficient as current increases and also as they heat up), go into thermal runaway and burn out. 9V across a single LED will generally burn it out fast enough to make it difficult to test. Was your friend perhaps adding a resistor to each set of LEDs in series and were they possibly different resistors?
If the 9V source is a battery, bear in mind a battery does not provide exactly its name voltage, has internal resistance and will change in performance if it heats up.
I use a "9V" battery pack with my breadboard (6 AA batteries in series) and with new batteries in it and no load connected it gives about 9.5V. If I connect a heavy load(extremely powerful COB LED) to it, even with new batteries the voltage of the pack can go as low as 6V. If you aren't reading the voltage during each experiment with a meter, you can't necessarily trust that a battery pack or voltage source is giving you exactly 9V.