A processor is the brain of a computer. Also referred to as the central processing unit (CPU), it is responsible for running all the computer's programs. A CPU with adequate power for your computing needs is absolutely essential for overall system performance – and maybe your sanity, as well.
These days, it's common to hear about multi-core processors. So the question becomes whether you need a single-core, dual-core, or quad-core processor. CPU speeds and multithreading capabilities can get complicated, but the main idea with cores is how many tasks a processor can run simultaneously. Having two cores is somewhat like having two processors in one. So the more cores you have, the easier it is for the processor to handle multiple computing tasks at once – and the faster it is. In fact, some programs are designed to utilize multiple cores at once, making them especially suitable for multi-core CPUs.
On the other hand, this extra speed may require more energy, thus affecting efficiency and, for laptops, battery life. Still, manufacturers are constantly pushing the envelope in terms of providing processing speed and low energy usage, so it's a good idea ask about the current tradeoffs between power usage and speed.
Intel i-Series processors represent the state of the art in terms of processing speed and multi-core availability. Don't get too hung up on the names, though. Since Intel is always improving their products in terms of speed, efficiency, and size, you'll hear models referred to as second- or third-generation. For simplicity's sake, these "generations" also go by codenames, such as Ivy Bridge or Haswell. As of this writing, Intel's third-generation Ivy Bridge processors are being replaced by fourth-generation Haswell processors.
But the names will keep changing. When shopping, find out what the newest technology is called and what it offers. Typically, retailers will offer the newest CPUs on their higher-end models, while continuing to offer slightly outdated (but still powerful) processors on less expensive models. If you need what the latest and greatest processors offer, you'll be able to get them. If not, you can save some money by staying with the last generation.
Besides Intel i-series processors (they manufacture other series as well), AMD is a leading manufacturer of CPUs. AMD processors often provide the performance needed for many tasks, but are generally less expensive.