Catching or Hatching a chicken
In general, you'll want to first build a pen to hold them. Single-height wooden fences (or a small cave) will suffice, but either way it's best to add an "entry lock": a fenced space with gates leading both to the pen and to outside. This will help prevent escapees—besides the obvious, if one of the gates is always closed, the chicken's pathfinding will never see an escape route to the outside.
The usual way to capture chickens is to hold seeds in your hand. Once the chickens notice you, they will follow you, and you can easily lead them into your pen. With care, chickens can even be led across water, as they will follow your boat. The alternative option is to collect Eggs and throw the eggs into your closed pen. There is only a 1 in 8 chance of spawning a chicken when you throw an Egg, so you should try to collect at least one stack (of 16). They will take some time to grow to adulthood, but once you have at least one adult chicken, it will start producing eggs... and with two or more adults, you can breed them with seeds.
 Setting up the farm
Alternatively you can follow one of the tutorials below, to create a farm that channels eggs to a single point. Most such will do the same for chicken meat, feathers, and even experience orbs as well.
One or more hoppers can be placed to collect the items (though not the orbs) and channel them to a chest, and the collection chest can even be hooked up to a dispenser to automatically hatch the chickens. However, note that leaving a such a full-automatic system running for too long will produce "Chickmageddon"—in sufficient numbers, the chickens will overflow any enclosure, and huge numbers will cause the game to lag badly.
 Trench Farm
Note: As of 1.5.1, this design will cause the majority of chickens in the pit to clip into the wall and die. It's retained because once certain bugs in the game are fixed, the design may work again.
The 14 Second Compact Egg Farm is a farm designed by Minecraftmaximizer for the Minecraft 1.5 release which takes only 14 seconds to build. It costs 8 logs of wood, 10 ingots of iron, two arbitrary blocks, and an optional ladder.
This farm is begun by digging a hole 3 deep, by 4 long, by one wide hole. The chests and hoppers are placed on the bottom (a double chest on one side, two hoppers feeding into it on the other.) The ladder can go over the chests. Two blocks then go over the hopper next to the chest, to keep the chickens in place. Then you can hatch chickens over the exposed hopper, and eventually collect eggs from the chest.
Because it has a volume of only twelve blocks, this farm is one of the most compact farms possible, especially with the inclusion of hoppers. A video demonstrating it and a schematic:
|14 Second Tutorial: Egg Farm with Hopper Collection System Video (view on YouTube)|
 Hopper Farm
The hopper egg farm is a simple contraption with chickens resting on a hopper and the eggs flowing down into a chest.
 Water Egg farms
Most current egg farms have the chickens supported by water, with their eggs falling through the water into a collection area below. The water can be supported by signs or ladders, which will keep it from flowing into the collection area.
For a fairly space efficient design: Build walls around a 2x2x2 column. The bottom two spaces are the collection area (make sure to leave a door), and the upper of those has 4 signs to support the water. The next two layers are a water pool with no flow, especially not downwards. It's probably best to make the whole pool of source blocks. The chickens will go in and above the water—there should be a 1 or 2 tall gap above the water for the chickens to breathe in. The walls around and above the water should be glass blocks, to keep the chickens from suffocating each other against the walls.
After this is constructed, eggs can be thrown directly up from the collection area. The chickens will float on the water and their eggs will drop to the floor for easy collection, where they can be thrown back to hatch more chickens. When meat or feathers are needed, a sword can be used to pick off chickens from below. A water flow can be placed in the collection area to bring the eggs to one block, but this makes throwing eggs and collecting meat or feathers more difficult.
Note: Currently, this design is afflicted by game bugs: there will be escapees, and some chicks will manage to fall through the water into the collection area.
The static water of the design above lets some eggs get stuck on signs. Expanding the pool area (the "Flowing Egg Farm") allows a water current to gather eggs to the center, and the inward flowing current helps prevent chickens from "phasing" through the walls, allowing far more chickens to be kept. Variations: This design can be "squared off", flowing to a central 2x2 hole, or it can simply be extended horizontally, perhaps with another water current carrying eggs down the "collection corridor". It need not be the full 18 blocks wide, either, as long as the collection area is under where the currents meet.
|Flowing Egg Farm Video Video (view on YouTube)|
|Another Large Egg Farm Video (view on YouTube)|
 11x11x6 Automatic farm
This farm will be surrounded on the surface by an 11×11 fence or (better) wall (put doors or gates at the middle of a side). There is a pillar and partial roof in the center, and the "egg room" dug 3 blocks deep beneath that. (The egg room and its pillar can be adapted to other farm layouts.) You will also want a tunnel leading to the egg room, with space to get at the chest and other devices (you will at least need to retrieve meat and feathers), and the switch to turn it on or off. The chickens are contained primarily by water, so the farm partly resists the current problems with chickens walking through walls and fences. The schematics are below; mostly they should be obvious, but the "help" link gives a key. The least obvious bit is that the gold and stone-brick blocks represent "any full block", but the blocks shown as gold must also be opaque, while stone-brick blocks can be opaque, transparent, or in some cases air.
The base machinery includes three droppers, a dispenser, three hoppers, a chest, and a couple of switches. Redstone costs include two redstone repeaters, two redstone torches, and six redstone dust. Making the works from scratch, will cost a minimum of 6 smooth stone, 15 iron, 29 cobblestone, 10 logs of wood (with some bits left over), 18 redstone dust, 3 string, and a slab. Also needed are 7 solid opaque blocks, and several that can be opaque or transparent. You may well want to add an extra chest in the egg room for ordinary storage.
The 9×9 floor inside the room will need 78 additional blocks or slabs (plus any you put under the walls). The pillar and roof will need at least two blocks (one can be a jack-o-lantern), and 11 more blocks or slabs. Optional extra: A trapdoor from the chicken floor to the egg room. (The water not only won't flow through the trapdoor, but will generally prevent chickens from slipping down there too.)
A fence would cost another 15 blocks of wood or so, but then any chickens that do get out of the water (and some will) can sit next to them and be stuck on the edge. Glass panes have the same issue, so the "upgrade" has to be to a 2-high wall of blocks. This will cost most of 80 blocks of stone and/or glass (or 20 logs for planks). Given creepers, it's much safer to make at least the floor and the bottom row of the wall out of blast resistant blocks: Any stone (sandstone should be converted to double-slabs) will do, as will brick or hardened clay, or even obsidian. This will minimize the mess if it does get blasted, and make it much easier to fix up. Making the top row out of glass blocks lets you see in and out of the farm, which helps avoid creeper blasts in the first place. You can also surround it with other protections such as a moat (which prevents creepers from damaging the blocks even if they do explode).
Once at least the floor and above-ground parts are built, you can put in the four water source blocks.
 Building it
Once the walls are set up, it is easiest to build the egg room from above. Make sure to offset the room so the input hopper is in the center of the floor, and light the egg room properly! When orienting the room, think about where you want the access tunnel to go. As shown, an access corridor leading to the lower left of the diagram allows getting at all the containers and both switches.
For first time builders, the hatcher proper consists of two droppers facing up, with a dispenser facing up on top of them. These are fed by the hoppers, with the chest providing extra storage, and driven by a 3-clock. The clock is on the right edge of the diagram, from the block with the lever southwards and downwards. That lever lets you disable the hatcher completely—place it and turn it on as soon as the clock is built, so you can build the rest without annoying clicking.
The despawn timer (upper edge of diagram) is a dropper facing down over pressure plate. It works by dropping an item onto the pressure plate, which will turn off the torch and enable the clock until the item despawns. The block in front of the pressure plate helps avoid accidentally picking up the item as you pass near, but if you go close enough you can still pick it up and cut off the timer.Once you've built and connected the despawn timer, you can turn the lever back off, as the inactive timer will keep the clock disabled. The despawn timer's dropper can be loaded with any disposable item, such as surplus seeds or eggs. The block in front of the pressure plate is just to make it a little harder to accidentally pick up the item—glass will let you see if the item is on target, or has fallen off the pressure plate.
Once the egg room is built and closed over, continue with the central pillar: Above the hopper, place a top slab (you'll need a temporary block to one side), then two blocks above that. (You can make the lower one a jack-o-lantern, for simple lighting.) From the top block of the pillar, extend a roof out over the dispenser and at least one square around it in every direction. Put a torch on the roof to avoid unfortunate monster spawns. Note that if you use slabs, you may get chicks on top of the roof. If you just have the minimum roof, they'll just fall into the water, but if you want to extend the roof to the edges, use non-transparent blocks to avoid escapees.
Note that the dispenser is purposely separated from the collection hopper/central pillar, to allow for the dispenser's variable aim. The slab (or other transparent block) between them is only needed if you add the optional chest, but if you do, an opaque block there will prevent the chest from being opened, which also prevents the hoppers from adding or removing eggs from it.
Last of all, place buckets of water in each corner; they will flow to the central pillar. Load up your chest with eggs and set it running (or lead in some chickens). Then let the eggs accumulate until you have enough for a full run (a dozen stacks in the chest). For a longer run (say, if you want to multiply from a few chickens), you may wish to disable the despawn timer (you can add a lever to the block for its output torch), but see the warnings below.
 Running the farm
The clock is normally disabled by either the inactive timer, or by the lever. With the clock disabled, incoming eggs etc. will fill first the bottom dropper, then the bottom hoppers, then the chest, and finally the intake hopper. This gives a total of 52 stacks storage, or 79 with the optional second (large) chest.
Now, 79 stacks of eggs will produce an average of 163 chickens, which may be enough to seriously lag the game when you are nearby. Worse, they will take over 15 minutes to feed through (because the hoppers are slower than the clock). And if you leave the hatcher running much longer than that, the first chickens will grow up and start laying eggs. At that point, you'll be facing exponential growth, limited only by the speed of the hoppers. If the hatcher is left running after the first generation grows up, the system will be producing "only" 2.6 chickens a minute at first, but if the game doesn't crash, it will eventually peak at 18 per minute, 363 per game day. If you don't mind risking this, you can skip the despawn timer forming the top two rows of the egg room.
The despawn timer and inverter will enable the clock for 5 minutes only, letting you hatch 500 eggs at a time (about 31 stacks, producing an average of 64 chickens). There is a bit of a trick here: Since the clock has a period of .6 seconds, 300 seconds gets you 500 cycles. But the clock and dispenser are faster than the hoppers feeding the dispenser: The hoppers alone could supply less than 375 eggs to the dispenser, but the eggs in the bottom dropper give just enough of a head start to cover a batch of 500.