What exactly is process technology, and why does it matter? For the most part, it doesn't for consumers who are considering buying a particular processor or a fully built system—the hardware's capabilities are much more important. But process technology does matter a lot in the long run, as it allows computing technology to both grow (in power) and shrink (in size and energy usage).
I suppose the first question to address is: What is a nanometer? The obvious answer is that a nanometer is one-billionth of a meter. To put that in perspective, a human hair is somewhere between 50,000 and 70,000 nanometers thick; and an atom is about 0.1 to 0.5 nanometer thick. In other words, features on today's chips are really small. In the technology world, we now use nanometers to measure the smallest features on an integrated circuit (a silicon chip).