Egg (Normally from a chicken), hat pin or knitting needle or nail or cocktail stick etc. Paint or dye. Vinegar.

Using a Hat pin / needle or similar pointy object make two holes, one in each side of the egg. When doing this it is best to either hold the egg in your hand or set it back in its carton to secure it. This limits the chance of breakage. The bigger the holes the easier the egg is to blow but the final decorated object will be less pretty. Some people make the holes big enough to fit beads inside!
Once your holes are made try to break up the yolk by stabbing and swirling your pointy object inside the egg.
Now your ready to blow. Holding the egg gently and securely over a bowl, blow through the smaller hole. Try to blow at a constant pressure. It will be hard and and will take a few tries, you may have to rest for a bit.

Once you have removed the inside of the egg you will need to wash it out. To do this fill a small bowl with a 1/4 vinegar to 3/4 water solution. Dip the egg into the bowl and let it half fill. Give the egg a good shake and once again blow out the solution (Don't worry it will be a lot easier this time). Repeat the cleaning process until all of the original egg is removed.

Allow the egg to dry out for a day or put the egg shell in the oven to dry it before decorating.

Paint, dye or decorate the egg to your own taste.

An egg at room temperature is easier to blow than a cold one.

There is a "One Hole" technique, where rather than blowing with your mouth you insert a syringe into the egg and inject air. In this case the hole must be at the bottom and big enough to allow the egg to come out, as well as having the syringe through it. You may use bleach instead of vinegar to clean the egg if using a syringe to blow.

Varnish the decorated egg to protect and strengthen it.

"Fabergé eggs" are probably the most famous decorated eggs. Designed by Peter Carl Fabergé who devised a technique to enamel the decorated egg.