Hatching chickens takes 21 days. The easiest way to hatch a batch of chicks is under a broody hen. Put six or a dozen eggs under her - depending on her size - sit her somewhere where she won't be disturbed by other hens trying to lay in her nest; and like magic, three weeks later, chicks should appear.
But what if you don't have a broody? Well, you might try borrowing one. Or you could borrow or buy an incubator.
Incubators come in all sizes, from ones that will take seven hen eggs upwards. The main choice that you need to make is whether you want a manual one or an automatic. If you choose a manual one, you will need to turn the eggs twice a day for the first 18 days of the three-week hatching period. If you choose an automatic one, this will turn them for you (and you will need to stop the auto-turn on Day 18) - the downside is that they tend to be more expensive. However, a lot of people start off with a manual and then upgrade to an auto-turn - so there are often second hand ones for sale on poultry websites or on eBay. You can also buy kits to make your own, or build one from scratch yourself.
When you receive your eggs, let them rest point down at room temperature for about 24 hours before you put them under the hen or in the incubator. It is also a good idea to run the incubator for 24 hours before you put the eggs in, in case you need to adjust the temperature. Each incubator is slightly different and you will need to read the instructions to see how much water to add (to keep the atmosphere slightly humid and mimic the underneath of a hen) and how to set it up.
If you are turning manually, mark a cross on one side of each egg and a circle on the other, in pencil. You will then be able to see clearly to turn the eggs.
If you want to, after 9 days, you can 'candle' the eggs to see how they are developing. This is a good idea because discarding the 'dead' ones reduces the risk of disease and makes more room in the incubator for your hatchlings. However, some eggs, particularly dark-coloured ones, are difficult to candle.
There are all sorts of 'light boxes' that you can buy to candle your eggs - but the most simple way is to hold a small but powerful torch to the 'fat' end of the egg, so all the light is focussed inside. You should then be able to see what is going on within the egg shell.
A day or so before they hatch, you might hear the chicks 'pipping' inside the eggs - starting to peck at the shells to get out. It might take 48 hours before all of them hatch - if you have enough space in the incubator, leave them in for a few hours until they dry off - they live off the remains of the yolk inside them for the first couple of days, so they don't need food or water. However tempting, DON'T keep opening the incubator to check on them! However, every few hours - four or five, say - you can open the incubator, take out and discard any broken shells and remove dry chicks and transfer them to your brooder.
Sometimes chicks DO get stuck inside their eggs and it is considered bad practice to help them out of their shells. However, I don't think I know anyone who has hatched their own eggs who hasn't been tempted to give a helping hand at some point or another. If you do do this, be very, very careful, don't do it too early and be aware that you might end up with a deformed or sickly chick - there is usually a reason that they can't get out themselves.
When you have moved all your chicks to the brooder, make sure that you clean your incubator thoroughly, ready for next time.