Traditional egg-farming
Egg farming is the process of collect a large number of eggs from chickens.
Catching a chicken
Capturing chickens to stock your farm can be the hardest part of the entire process. To capture a chicken, you will need a few fences. You may need only four, but since you will fail a few times, you may wish to bring more. First, find a chicken. Simply walk around, but don't forget where your base is. Next, wait untill it is not in water, and is on flat ground. Then simply wait untill it stops moving, and rapidly build a wall around it. Dirt can be used, but due to the time it takes to place two blocks of dirt on top of each other, fences are recomended. Now, either, take its eggs and start a chicken farm somewhere else, or make it around your captured chicken.
Setting up the farm
You can farm eggs the traditional way, where you have to run around and collect eggs all the time.
Joghurts egg-farm
Or you can for example build an automatic egg-collecting farm by using water to flush the eggs out of the coop.
Bottom layer of Joghurts egg-farm
I tried several designs and one that works great is the one from the second screenshot: Just dig a square hole 11 by 11 blocks and one block deep, then dig one more layer in a circle shape. You'll need an aditional one-block-hole in the center where the water (and the eggs) will flush out.
Now you need to build the "isle" to prevent the chicken from also being flushed out and to give them some ground to lay eggs that can hatch on to slowly expand the total population of chicken on your farm.
Connect the center drain hole to your system from below (F3 is your friend here), place water in the corners and you're good to go.