Coldness is just the absence of heat, so refrigerators don't really "produce" cold as much as they remove heat. They do this with a chemical refrigerant, which is housed in a continuous tube that runs from the outside of the appliance to the inside and back out again. In basic terms, when the refrigerant is inside the appliance, it absorbs heat from the air there. It then carries the heat out and releases it into the surrounding room. In turn, the contents of the refrigerator lose their heat and become cold.
A typical modern refrigerator has two compartments: the freezer and the refrigerator. The transfer of heat between the air and the refrigerant occurs only in the freezer compartment. In other words, it's in the freezer that the air first "gets cold." Vents between the freezer and the refrigerator control the flow of air between the two compartments, and that's how the refrigerator gets cold.
The air diffuser is the mechanism that handles air transfer between the freezer and refrigerator compartments. The diffuser has a temperature sensor, which you control with the refrigerator thermostat -- the knob or lever you use to set the temperature inside the refrigerator compartment. When the diffuser senses that the air in the refrigerator has gotten too warm, it opens the vents between the two compartments. "Warmer" air from the refrigerator flows through the vents into the freezer, where it can be cooled by the refrigerant; colder air flows from the freezer into the refrigerator, where it lowers the temperature. The diffuser usually has a fan to help the exchange of air.
If neither the freezer nor the refrigerator is getting cold, the problem likely lies in the system that circulates refrigerant. If the freezer is getting cold enough to freeze food but the refrigerator is too warm no matter how low you set the temperature, the problem will often be a bad diffuser. The diffuser isn't exchanging enough air -- or any air at all -- between the two compartments, so the refrigerator isn't cooling.