What parts are strictly necessary for a 3D printer? I've built the mechanics of my 3D printer myself, because I need to print parts that are really huge, (and for budget reasons). So, I already have the 3D movement functionality.

But what I need now, is the printing mechanism itself. I've been reading a lot, but it became clear to me that things are more complicated than I thought. 

Let's skip mechanics and software, I'm just interested in how the print head works. Can somebody explain me that?

To be honest, I was so naive that I thought that I just had to buy one part with one data wire (print/noprint) and the 5&nbsp;V/GND wires. But it came to my intension that things are way more complicated. 

For example, these RepRap printers have some kind of air tube attached to the print head. I'm not sure what that's all about, is it cooling?

Perhaps I'm always reading the wrong manuals (i.e. the more advanced ones). Can somebody enlighten me or point me to a good starting point?
You will certainly find that the print functionality of a 3d printer is a bit more complex than you suggest. The mechanical portions include a means to push the filament into a heated nozzle as well as the software portion to regulate the speed of the filament movement. You haven't referenced the heater cartridge and temperature sensor, but you will discover that aspect soon enough.

The "air tube" you think you've seen is likely called a bowden tube. Such designs permit lighter weight print heads, which is beneficial for speed, acceleration and precision, but has complications with respect to compression of the filament as well as retraction considerations. Non-bowden print heads will have the extruder motor as part of the moving assembly, with the drive wheels very close to the nozzle opening. This allows for flexible filament and more precise control of the filament feed. 

Either design has compromises, so one must determine priorities for the design.

Cooling is also a factor. The heater cartridge is designed to heat the nozzle to a specific temperature for the type of filament used, but also requires a means to keep the heat from traveling to the portion of filament not in the nozzle. You'll discover terms such as heat break, referring to narrow threaded portion connecting the nozzle assembly to the heat sink. There will also be a cooling fan to blow air over the heat sink and very often a cooling fan to cool the filament as it exits the nozzle and attaches to the model being printed.

You suggest to ignore the mechanics and software, but it's important to be aware of both when considering the principles of the print head assembly.

Simplified, filament enters bowden tube then into heat sink, pushed by extruder motor (or) filament is pushed into heat sink by extruder motor. Filament travels through heat break, gets melted in heater block and exits nozzle. Sheesh, that's way too simple.