The way that they work is the power supply provides the necessary voltages to power the printer. The main logic board controls all of the functions of the printer plus translates the print job sent to it so that it can print. The printhead has solenoids that push the pins outward to the ribbon, the main logic board controls which pins are pushed. The carriage is the assembly that has the print head on it that moves back and forth. It also has a sensor or device that tells the main logic board at what position the carriage is at.
Some printer have a LED sensor, some have pushbuttons, some have a belt.
There is a servo motor that drives the carriage called the printhead motor. This motor moves in both directions as directed by the main logic board. Usually it moves the carriage by a belt or gear assembly. Also there may be a motor or gear assembly on the printer that turns the ribbon. Some printers have a big black roller called the Platen this is a rubber tube on a shaft that provides friction so the paper can be moved properly to print.