A computer's processor determines how efficiently it can run programs, multi-task and basically do everything we expect of modern computers. Processors get faster and more efficient every year. Most Windows-based computers run on Intel's processors; smaller ultraportables, such as Apple's thin MacBook Air, run on ultra low voltage processors that draw less power than some of Apple's other chips. Quad-core chips deliver more powerful performance, but even dual-core processors are up to the task of playing 1080p video and running system-intensive programs like Photoshop.
The graphics processor, or GPU, is important when it comes to playing HD video and running games. Many laptops use integrated graphics rather than dedicated graphics chips. These are less powerful (and also less battery intensive), but powerful enough to decode 1080p video. Dedicated graphics chips are important for playing video games, but for the average computer user, the two components on the next page -- hard drive storage memory -- are more important.