The Chris Houlihan Room
FIRST (EASIEST) METHOD:
Now, you're going to have to go to the secret entrance to the Hyrule castle (the one behind the bush you go to at the very beginning of the game) AS FAST AS POSSIBLE. The trick will only work if you're fast, which is why you need the Pegasus Boots. Try to storm your way through the screens.
Here is the exact path [note that any path is theorically good as long as you reach your goal quickly enough, but this one works fine for most]: go down to leave the Sanctuary, down past the cemetary, then left. In the new screen, go down as soon as possible, and you will enter a difficult area with big trees, rocks, bushes, and enemies. Proceed with caution here, as it's easy to waste time if you're not careful enough. Go right of that screen. You are now at the Hyrule castle. Go down, then right, and go past the main entrance without entering. Keep going right until you're at the eastern side of the castle, then go up. You will find the bush leading to the secret entrance. Get rid of it and drop down the hole, and IF you were fast enough, you will enter the Chris Houlihan room.
It's the same principle, but instead of starting from a saved game at the Sanctuary, from anywhere in the game just go to the Kakariko village, taking as long as you wish. There, you have to place yourself at the top of the dirt path leading to the fortune teller screen. The race against the clock starts now: run up to the fortune teller screen and go to the secret passage behind the bush just as in the first method (go right and then it's all the same directions), always running as fast as possible.
There is a rumour the egg also appears when the game somehow malfunctions (instead of giving you an error message, it warps you to the Chris Houlihan room), which could explain why the many reported ways to enter the room are so strange and different.
Note: This text is taken from The Easter Egg Archive website, http://www.eeggs.com/items/26809.html, used without permission. But, hey, I just included a link to their site right there, so if they get all pissy about it then they're just freakin' idiots. Free publicity is a good thing, man!