A trick I often use is to puncture the fat end of the raw eggshells with a pushpin, easily done by holding the pushpin pin-side-up on a flat surface and GENTLY, but with purpose, push the fat/rounded end of the eggshell onto the pin, to make a hole. Then place in a pot of cold water, with water about 1" above eggs. When the water boils, immediately move the pot off the heat and let sit for 13 minutes (for hard boiled), then rinse under cold water.

Some people add a teaspoon of salt to the boiling water.

Another trick I've heard of is to add a tablespoon of vegetable oil to the boiling water. Egg shells are porous, and absorb the oil. Beyond that I don't really know why it works, but I've heard it does.

For musical entertainment while your eggs are boiling, this site is fun: http://www.eggwatchers.com/

The secret to peeling the eggs is not so much as in the boiling but in the immersion in cold water once off the heat. I always tip out whatever hot water is left in the pan, with the cold tap running and immediately cover the eggs with cold water. I let it sit for a few seconds and then I tip that water out and replace with fresh cold water. It works 98% of the time.

also it helps to plunge into cold water when finished cooking.

Older eggs are easier to peel and fresher ones are harder.