Chickens Hatching Eggs
You don't need any fancy incubation
equipment if you've got a broody hen...
you just need some fertilized eggs.
If you're ready to try hatching chicks
the old fashioned way,
here's a great place to start...
Broody Hens
There are times, as a chicken owner, you might encounter a broody hen. Essentially, a hen goes broody when she gets swept up in the urge to be a mother, and decides to sit on a nest of eggs (her own eggs as well as other hens) and hatch them.
Some people don't want a broody hen. They want a hen who will lay eggs without getting sidetracked by raising a family. Allowing a broody hen to run the course of nature means lost egg production for the 21 days the hen sits on her clutch of eggs as well as some time afterwards when she's raising the chicks.
Broodiness has been breed out of many of the production hens these days to make them more effective at laying eggs. If you're one of those people who would rather not deal with a broody hen, choose breeds of chickens that are less likely to brood, such as Leghorns, Sex-links, Rhode Island Reds (or Whites) and others. And stay away from Silkies, Cochins, Brahams, Orpingtons and others.
How to Let Nature Run its Course
If, however, you do have a broody hen and you'd like to allow her to sit on a clutch of eggs, you first need to get your hands on some fertilized chicken eggs. Since most cities don't allow roosters, chances are, you'll have to get some eggs from somewhere else. There are several online hatcheries that sell fertilized chicken eggs where you can buy them. You could also check locally to see if there are nearby farms that have fertilized eggs they want to sell.
It's good to act upon this as soon as you see your hen refusing to budge from her nest, as that means her hormones are primed and ready to roll. After obtaining the eggs you'll give her to sit on, it's best to wait until it's dark outside to give her the eggs. Gently remove any eggs that she's already sitting on, and replace them with the new fertilized eggs. Often you can place other kinds of eggs under your hen and she'll hatch them out just fine (other breeds of chickens, ducks, quail, etc.)
(For a smaller breed of hen, the maximum number of eggs she can effectively sit on is about six. For larger breeds, you can try up to 12.)
If it's possible, separate the hen and her clutch of eggs from the rest of the chickens to protect her from getting picked on and to keep her eggs from getting broken by the other hens. Again, if she's already established a nesting space and you feel it's in a precarious place for her, try moving her and her nest at night.
Taking Care of Mama
A hen who is busy trying to hatch a clutch of eggs will rarely leave her nest. During this time, make sure there's food and water nearby for her to eat. She won't eat as much as usual, but that's because she's not moving around much.
Also, watch to make sure the other chickens aren't picking on her (if you haven't already separated her out from the rest of the flock).
When the Babies Arrive
After 21 days, the eggs will begin to hatch. Although most eggs hatch at 21 days, there is a 36-hour window for hatching. During hatching time, don't disturb her. If all the eggs haven't hatched at the same time, wait a day or two longer and then remove all the unhatched eggs. If some of the eggs hatch earlier, the hen might abandon the unhatched eggs to care for the chicks she's already got.

Once the chicks are born, place chick starter feed and small water containers near the nest. The new mama will take it from there. She will teach her chicks where and how to eat, keep them warm, and protect them, even to the cost of her own life.
If you want to separate the hen from her chicks, you can do that at any time. However, it's best to wait until they're 4-6 weeks old and fully feathered out. If you remove the chicks before then, they'll need to be put in a brooder with a heat lamp until they can keep warm themselves (after they're fully covered with feathers).
Pros and Cons of Hens Hatching Chicks
The pros of using a broody hen to hatch chicks is that they do it best. It's the way nature intended it. And (generally), most breeds of chickens that tend to go broody also make good mothers. And if you have a broody hen to begin with, it's a good way to get her over it--just let her do what nature is telling her she wants to do: Raise babies.
Having a chicken raise her own chicks is a good way to add to your flock without the issues of introducing new chickens to the ones you already have, which can often be stressful on both new chickens as well as chicken owners.
It's also a great learning experience for all involved. How cool is it to participate in the lifecycle of going from eggs to chicks to chickens? Not only that, but from egg to chick is only 21 days, so you don't even have to wait very long for the experience.
One of the cons to letting nature take over, is that you've got a 50/50 chance of getting a roosters. For those of you raising chickens in the city, this might be an issue, as often roosters aren't allowed.
Also, there's a chance that not all your eggs will hatch (so if you're aiming for a certain number of new chicks, you might not get them).
Additionally, if you're getting eggs from a local farm, you're less likely to be able to choose exactly the kinds of chicks you'll be hatching.
Another slight con for city folks is that most small backyard flocks are tame, especially those chickens who were obtained as chicks and raised by hand. When baby chicks are raised and nurtured by humans, they get accustomed to being handled. However, a hen will do her best to keep you away from her chicks, so hen-raised chicks aren't nearly as tame as their human-raised counterparts.
All in all, though, I'd encourage you to allow your broody hen to sit on some eggs at least once, just for the experience of it. I don't think you'll be disappointed.
More information that might interest you:
Before you Buy Baby Chickens
Taking Care of Baby Chicks
Getting Started with Chickens
Chicken Raising and Care
Basic Chicken Coop 101 and Basic Chicken Coop 102
Free Chicken Coop Plans
How to Make an Automatic Chicken Waterer
Weird Chicken Facts
