Ways to Hatch
Whether you can hatch naturally or only in an incubator depends on the breed you have selected (and oftentimes the age, as well). Some breeds have had the broodiness completely removed over the years, while others will happily settle down on a nest a month or two after they start laying.
Even breeds that don't have a strong broody instinct can become good parents over time. As a hen ages, she slowly becomes more interested in settling down and raising a family. Usually 2-3 year old hens will consider parenthood, although they may not be the best of parents. It is up to you to decide whether to take the chance that an older hen will sit on the nest for the whole three weeks (and then will still have enough of their innate mothering instincts to tend to the chicks if you leave them with her).
Assuming you have a broody hen, you have three options for increasing the size of your flock through hatching. The first is the most common: to take the eggs and put them in an incubator, then raise them in a brooder. The second option is to allow the hen to incubate the eggs, then take the chicks away to keep them safe from any accidents. The last option is to have the hen incubate and raise the chicks herself.
Artificially incubating and brooding chicks is the most common method used by urban farmers. However, hatching eggs is an artform that has to be perfected over time. You also need to buy an incubator, which can become pricy as you buy the add-ons to increase the hatch rate (such as an egg turner, a fan, and an automatic waterer to keep the humidity at the right level). But after the initial expense and a lot of trial and error, incubating out your own chicks allows you to to hatch the eggs while your hen is still laying and cuts down on the need of having older hens around that don't produce very well. This also allows you to hatch out a larger number of chicks, thereby raising your next generation all at once instead of a handful here and there.
By allowing the hen to naturally incubate her own eggs, then remove the chicks when they hatch, you cut down on her egg production for a 4-8 weeks (the three weeks she is on the nest then some time to recover from her ordeal), but get rid of the need of buying and maintaining an incubator. When the chicks hatch you can make sure they stay safe by keeping them in a separate brooder under the perfect conditions. This also makes it easier to feed the chicks, since laying hens prefer chick starter and grower over their own laying pellets and will do what they can to get to it. It is a good idea to have two or three hens sitting on eggs at once so if one abandons her nest you can transfer the eggs to another nest.
If you decide to let the hen hatch and raise her own chicks you are removing her from egg production for 8-12 weeks and making special arrangements for the babies: making sure they aren't bullied by any other adults, that they are safe from wild rats, their food isn't stolen, and that their mother doesn't get annoyed and "wean" them too early. Usually it's best to have a brooder cage set up next to the pen (or even inside the pen) so that the chickens can still see one another without getting too close. This will make reintroducing the hen and her offspring less hectic. If you decide to go this route, your chicks, few that they might be, will be more "world wise" than artificially raised babies. Chicks learn a lot from their mother, and from the rest of the flock. This gives them a headstart on other babies - a headstart that they add to when they are old enough to raise their own family.
Artificial Incubation
If you decide to incubate the eggs yourself, touch them as little as possible (it isn't a bad idea to pick them up with a paper towel). The oil on your hands will clog the pores, reducing your overall hatch rate. In hot or cold weather, collect the eggs a few times a day to keep them from becoming too chilled or too warm. If the internal temperature of the egg gets below 45 F it probably won't hatch. Eggs that are above 85 F may start to grow before being put in the incubator, also reducing the hatch rate.
After you have gathered your eggs, keep them in a cool place with high humidity: anywhere from 55-65 F and around 70% humidity is ideal. Turn them once a day to keep the yolk from settling at the bottom and sticking to the shell. Once you have gathered enough eggs (no more than a week's worth: after the first seven days of holding the hatch rate suffers), place them all in the incubator together with the tips pointing downward and the air bubble (which can be seen if you shine a light through the shell), pointing to the sky. Any dirty eggs should be washed in water warmer than the egg and can be hatched if absolutely necessary, but are probably better to just put in the fridge for eating. When you wash the eggs you remove a protective coating that reduces the chances of them hatching.
In a still air incubator the temperature should be kept at 101 F with a 60% humidity. Forced air incubators (those with fans installed) should be set for 99.5 F with 60% humidity. Eggs need to be turned at least four times a day. If you plan on hatching a full incubator's worth (42-46) then I strongly recommend an egg turner to save time.
On the 18th day, turn off the egg turner or stop hand turning the eggs. At this time the chicks are shuffling around inside the shell and positioning themselves for the hatch. Turning the eggs at this point will only make their maneuvering more difficult.
On the 21st day the chicks should hatch. Some hatchers increase the humidity by 5-10% at this time to help loosen the membrane and make hatching easier. Others say that the increased humidity causes the chicks to hatch too quickly before they have completely absorbed their yolk, and prefer a "dry hatch" (ie. maintaining the 60% humidity until the end).
Once they begin hatching, resist the urge to open up the incubator to inspect them, lend a hand, or remove a fluffed up chick. Each time you open the incubator you are allowing cold air in and causing the humidity levels to drop, which can kill the chicks that are still trying to break free. Be patient and allow the incubator to stay closed for the next 12-24 hours. The chicks that have already hatched are fine without food or water for the first 48-72 hours and will be perfectly fine with the rest. Also, their peeping will encourage the others to hatch out.
Raising
Once you are sure all of the eggs that will hatch have done so, move the dried off chicks to a brooder that has been set at 95 F using a heat lamp with a 100-250 watt bulb. Brooder temperatures can fluxuate, so spare the few dollars it costs to get a thermometer or two set up inside.
In addition to the thermometers, you can see how hot or cold your chicks are by looking at them. If they are all huddled under the heat lamp then they are too cold. If they are as far away from the light as possible, or they are panting with their mouth's open, they are too warm. You want your chicks to be bouncing around inside the brooder all over the place.
Give the chicks a medicated starter feed to get them off on the right start. If you are raising some chicks for meat, they can be started on a broiler feed instead. To further bolster their immune system, add a little honey to their water (and keep the water warm for the first few days) and crumble hardboiled yolks on the ground. Honey has natural antibacterial properties and the yolk is packed full of nutrients for the babies.
Each week the temperature inside the brooder can be reduced by 5 F. Usually by the third or fourth week a heat lamp is no longer necessary so long as the temperature is at least 70 F.
Breeding
Most hens start laying at 6-7 months, with some hybrids and good laying strains starting as soon as 20 weeks. Meat breeds can take eight or more months to begin production, especially if they mature over the winter months. To get fertile eggs that can be hatched you have to have the right number of roosters for that specific breed. For example, heavier breeds tend to be able to mate with fewer hens than their lighter, more agile bretheren, so while a Jersey Giant rooster may be needed for every 6-8 hens and a dual-purpose breed happily services 8-10, a spritly Leghorn could handle as many as a dozen. Having too many roosters in the pen is also counter productive since the squabbling and excessive mating may stress them. If you have a mixed flock then use the rooster breed or body type (layer, dual-purpose, or heavy/meat) to determine how many males is enough.
For maximum egg production over the fall and winter months put a regular or florecent bulb on a little $5.00 timer to bring the artificial light up to 14-16 hours. While this is used by many people without any noticable adverse effects, some people say that not letting a hen rest over the winter months may lower the number of years she will lay well, or even the number of years she may live. Currently I have found zero evidence for or against keeping it "summer" throughout the year.
Regardless of whether you put light on your hens they will go into a molt at the age of 18 months, then once a year for 8-12 weeks from then on. During the molt the hens will drop all of their old feathers as new ones grow in and the stress on their systems will keep them from laying. During this time, I like to feed protein-rich foods since it seems to help with feather growth (feathers being almost pure protein).
When the rooster mates with a hen, she can produce fertile eggs for up to two weeks - just from that one day together (and there have been some reports of up to three weeks). If you have more than one rooster, or have recently purchased a hen, keep this in mind if you are wanting to have chicks that are out of a particular male.