Fresh eggs (less than 3 days old) are harder to peel because the white membrane is just not mature enough. You should buy eggs for hard-cooking at least a week ahead of time
A simple test in water will answer the freshness question for you. Place the egg in a bowl of water; if it lays on its side, it is very fresh. As it ages, the air pocket inside the egg grows, which buoys the egg up so it stands on one end. If the egg floats to the top, it is ready for the trash.

Basic Cooking:
Put in a pot big enough to accommodate the eggs in a single layer with 1 in of water over the top.
Bring to a boil. Simmer just below a boil for:

extra large	15 min
large	14 min
small and medium	12 min

Plunge them into cold water for 10 minutes to stop cooking.

Optional:
Note: Some instructions call for removing them from the stove as soon as the water boils or in 5 minutes, covering them with a lid, and letting them sit for 17 - 20 minutes, depending on size, while the water cools. Then place them in ice water for 10 minutes to make them easier to peal.

Some put them back in boiling water for 30 sec. just before pealing.
The cooling process allows the inside of the egg to shrink faster than the shell (which is less prone to expansion and contraction) and a layer of steam fills the space and therefore the egg will not be as tightly adhered.

Overcooking will result in a green color around the outside of the yolk that results from iron in the yolk reacting with sulfur in the egg white. It's not an appetizing hue, and it carries a rather sulfurous odor with it.
The ice water trick can be used to avoid this also.

Add some vinegar to the water to make the shell softer and easier to peal. It will dissolve some of the calcium carbonate in the shell. It will also help any whites that do escape from cracked shells to set more quickly.

How to tell if eggs are hare boiled:
A quick test to ensure that your eggs are hardboiled: When eggs have cooled, spin them on a hard surface (just like you would spin a top). If the eggs spins quickly without taking off or flying off in one direction, the egg is hard boiled and finished. Undercooked eggs (or uncooked eggs) will have a wobbly and unsteady spin.
Or, if you stop the egg quickly then let it go the fresh egg will start spinning again.