The processor also known as CPU (Central Processing Unit) is the brain of a computer. The faster the processor, the faster it will execute instructions and run your programs. The leading Processor manufactures are Intel and AMD. Whichever manufacturer you decide to choose you will have to make sure that you purchase a compatible motherboard.
Nowadays Intel and AMD have stopped the GHz race and concentrated on how many cores they can fit in each CPU. So having a 3 GHz single core CPU is not better that having 2.4 GHz Dual Core CPU. Nowadays a CPU is judged on how many cores it has. The more cores it has the better it is.
Before buying your processor consider what the system is going to be used for. If you are going to use your system for word-processing, browsing the web or other basic tasks, it may be sufficient for you to use an Intel Celeron, AMD Sempron or an AMD Athlon 64 processor, which are available at a very competitive price.
If the system is going to be used for spreadsheet, databases, graphics and playing some moderate video games then you should consider an Intel Core 2 Duo or an AMD Athlon 64 X2 processor.
Intel Core 2 Duo and Athlon 64 X2 are both Dual Core. Which means they have two CPU cores in one Chip. Each core works separately hence the CPU can process two intensive tasks at the same time without slowing down the system. For example you might be running a virus checker in the background while playing games without slowing your system down. They are also 64 Bit and support 64 bit Operating System.
If the system is going to be used as an advanced workstation for doing work such as Video Editing, CAD, or playing the latest 3D video games or other CPU intensive tasks then you will need a higher end quad-core or multi-core processor such as AMD Phenom X4, Intel Core 2 Quad or Intel i7.
AMD Phenom X4 and Intel Core 2 Quad has four cores. Intel i7 has 4 cores but each core can process two threads so your operating system will treat it as 8 different processor. All of these modern processors are 64 bit which means they can execute 64 bit instructions and run 64 bit operating systems. They are all very fast and are capable of handling most of your computing needs.