What is a CPU, the long answer. Think of it as a whole bunch of switches. Processors used to just have one core, these days, we demand so much speed from our computers the manufactures are building dual and quad core chips with current talk being about six core chips. Each core, is like another CPU.
If you are not interested in overclocking, the fan that comes with your computer CPU will be fine. If you know at some point you would like to try overclocking, or would just like to have to most stable system possible, get an aftermarket heatsink and fan.
It used to be easy to compare different processors because you could just choose the highest number was the fastest chip. That is no longer the case, don't rely on the Mhz or Ghz number to know which one is faster. Comparing a benchmark is the only reliable way to know where a processor rates as far as speed goes.
Make sure the processor is supported by your motherboard before buying. The easiest way to do this is to go to the manufacturer of the motherboards website.
Picking out a processor is a job I usually find easy after I have my motherboard picked out. I simply pick out the chip that ranks the highest on a benchmark that lets me stay in my budget of 40% for motherboard and processor. Either AMD or Intel makes a good choice for a processor.
If you are on a tight budget you can probably get more speed by going with an AMD chip. Remember to always check the benchmarks and the reviews about a processor before you buy one. Go to the benchmarks and reviews section on the How To Build A Computer page to learn more.