To start, there are some basic things you should know. The first is called TDP. TDP stands for Thermal Design Power, and is listed in the online specifications of your CPU. For example, my FX8350 has a TDP of 125W. Any cooler I purchase then has to be able to cool a 125W TDP or higher.
The second thing you should know is that heat WILL reck your CPU. Almost all CPU manufacturers have a listed maximum temperature for that processor (CPU).
The third nugget of information is that higher clock speeds and voltages equal more and more heat. If you plan on overclocking, your TDP will go up, and you will no longer know exactly what it is.
This next section will introduce some cooling methods. Hopefully, it will help you choose the one that is right for you.
Stock Air Cooling
Stock air coolers are what come with your CPU. They are usually pretty loud and allow minimal over clocking. Some, like AMD's eight core cooler use heat pipes that pull heat away and up into the fins of the cooler, where the heat is dissipated. In my opinion, heat pipes work better at pulling the heat away than a solid copper plate, like Intel uses. If you don't care about noise, go into your BIOS and set your CPU fans minimum speed up a little higher.
AMD's stock 8 core cooler. You can see the heat pipes that attach to the baseplate. This cooler uses two.
Intel's stock air cooler. This cooler has a solid copper baseplate, which I believe to be less effective.
Aftermarket Air Cooling
Aftermarket air can be quite efficient and quiet. However, many are quite large. Almost all aftermarket air coolers that are worth buying will use 3-4 heat pipes. The lower the TDP your air cooler is rated for, the faster and louder it will be. For example, if you use a 125w processor on a 125w cooler, your cooler will run louder all the time, in order to keep your CPU cool.
As you can see, this air cooler is quite large, utilizes many heat pipes and can sometimes cause issues by not fitting in your case. If your RAM has large heat sinks, you may want to make sure you have free return shipping on your cooler.
Liquid cooling looks cool, and does do a good job at keeping that CPU cool too. Liquid cooling can come in several configurations. The lower end setups may not be any better than an aftermarket air setup. This first picture is of a complete setup. Completes come pre built, and will fit on a case fan hole. I use an older Corsair H50 with an extra fan. The radiator can get quite hot, but as long as the temps are ok, you should not have an issue. The point of liquid cooling is to have the liquid absorb the heat of your CPU and have the radiator cool that fluid.
The other route with liquid cooling is to build a custom system, but that is not something for beginners. If you are one of the unfortunate ones, your cooler could spring a leak and kill your computer.
The benefit to these coolers is that you can select the exact component to cool. There are water blocks for GPUs, CPUs and chipsets.
For those extreme overclockers, use liquid nitrogen.
To conclude, as a general rule, your CPU should never NEVER be above 200 degrees F. Know your max, know you cooler and know your tasks.