Airflow, literally, is the flow of air around, into, or through an object. In the context of your question, airflow refers to the movement of the air into your case, inside your case, and finally out of it.
Airflow is important because air is a carrier of the heat energy dissipated by various components that make up your computer. Lots of things generate heat inside your PC: Your CPU, GPU(s), and motherboard chipsets are the obvious usual suspects. Heat is basically radiated from these hot bits and into the air inside your computer's case.
So how does airflow factor into the cooling of your computer? Well, there are basically two schools of air cooling thought here: Positive pressure and so-called negative pressure. Positive pressure means that there is more air pressure coming into the computer's case than there is exiting it. This is accomplished by having a larger volume of air entering your case than the volume of air exiting. How is this done? Have more fans actively pulling air into your case than pushing it out. Typically, this is by having powerful fans as intakes, and having no fans for exhaust.
Negative pressure, on the other hand, means having a greater volume of air leaving the case than entering it. You do this by having powerful fans actively sucking air out of your case, thereby exhausting the heated air. You may notice that this is the exact opposite of the positive pressure concept. To my knowledge, this method is more popular and more effective than a positive pressure case.
Airflow is also influenced greatly by temperature. As you know, hot air rises. Therefore, it would help your case's airflow if you took advantage of this physical fact. This is why you see most cases having their exhaust fans high and to the rear of the case; some cases even have "blowholes." This is also why most cases have intakes at the lower portion of the case's front. Cold air, after all, is heavier, and is therefore closer to the ground.
The air inside a conventional case, then, moves from low down in front of your case, gets heated by the various heat-producing parts inside, then rises towards the back and top. Knowing this, strategize accordingly how you would place your fans and at what speeds they'd run. For a negative pressure case, low speed/pressure fans go up front, and higher speed/pressure fans go to the back and higher up. Always strive for having more pressure leaving your case than entering it, in my opinion.
Lastly, how do you improve airflow? Air moves best when there's nothing in its path. This is why people take so much effort in managing their cables. Minimizing obstructions in the air stream is another key to good airflow. Cables and wires create aerodynamic drag, which interferes with the movement of air to the hot bits inside. Turbulent air is a poor dissipator of heat, as well.
Sorry for the rather lengthy post. I hope this helps, though.
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