If a chicken lays ONE egg a day, and has several baby chicks, does the rooster impregnate her for every one of those chicks, once everyday? Or, does he have to do his thing just once, and that one will last for a while? And how does she know to stop having eggs or chicks?
What usually happens:
--The rooster typically mates with every hen, several times each day.
--A hen's body can store sperm, so she really only needs one mating every week or so, and that is enough for every egg to be fertile. This also means if you have a rooster and he dies, you can typically collect eggs for a week or more and hatch them.
--When a hen needs to lay an egg (about once every day or two), she will go find a nice nest to lay it in. She especially likes to find a nest that already has eggs in it. After she lays her egg, she goes away and ignores the nest until the next time she needs to lay an egg. It is common for several hens to lay eggs in the same nest. So the nest can get a big pile of eggs quite quickly, unless you collect them regularly.
--At some point, a hen goes broody. She sits on the nest full of eggs, and her body heat causes an embryo to start growing in every fertile egg. She will get off the nest for a short time each day to eat and drink (maybe a few minutes, sometimes up to an hour), but then she gets back on the nest. She will usually fluff up her feathers and make loud noises if you disturb her.
--After the hen has been sitting for three weeks, the chicks should hatch. The hen will keep sitting for another day or two, then she will lead the chicks away from the nest and show them what to eat and drink. When the chicks are cold, they peep loudly, and the hen will sit down and let the chicks warm up underneath her. If other chickens come over to see the chicks, the mother hen will chase them away, to make sure her babies are safe.
--At some point, the hen will decide that her babies are grown up enough, and will leave them alone. She rejoins the main flock, and at some point she starts laying eggs again. She might abandon the chicks when they are only 3-4 weeks old, or she might keep mothering them for much longer than that.
That is what usually happens when everything goes well.
Common variations:
--Some hens never go broody. If you want their eggs to hatch, you have to put the eggs in an incubator, or put the eggs under a different hen that did go broody. Because a hen stops laying eggs when she goes broody, the best layers are usually the ones that do not go broody.
--Of hens that do go broody, some are not good mothers. Some hens stop too soon, before the eggs hatch. Or they kill the chicks who hatch (That fluffy thing must be attacking the eggs! Kill it!) Or they completely ignore the chicks who hatch. Or they are good mothers to some chicks but reject some other chicks (some hens are fussy about chicks that are a different color than the others, but many other hens are happy to mother chicks of all colors.)
--If a broody hen is sitting on a nest, other hens may shove in and lay more eggs in the nest. The shoving can break eggs. The new eggs will start to develop, because they are now under a broody hen, but they got a late start. So when the first chicks hatch and the hen leaves the nest, those later eggs stay in the nest and get cold and die.
--If the eggs are infertile, they will never hatch. Or if they were fertile, but they got too cold or too hot before the hen started sitting, they might not develop either. In that case, the hen will usually sit for three weeks, and a bit longer, but as some point she will usually give up.
--Hens do lose weight when they are broody. Up to a certain point this is fine, and may even be good for them, but of course taking it to extremes is bad for them. Losing weight is partly because her reproductive organs become much smaller, and her body re-absorbs the egg yolks she was making. A hen usually has quite a bit of stored fat before she goes broody, so she will also use this up. Using up some of the stored fat is probably good for her. I have read a few stories of hens that starved to death because they sat for a long time and did not eat enouch, but those are fairly rare. I have personally seen a hen that sat for over 6 weeks before she quit, but she did not seem to have any problems. That would be something that varies from one hen to another.
--A hen does not care whether she sits on eggs that she laid, or eggs that another hen laid, or fake eggs, or anything vaguely egg-shaped. So if you want specific eggs to hatch, you can give them to any hen that is broody, even if she is not really the mother.
--Eggs can usually be stored for at least a week between when they are laid and when they start being incubated (under a broody hen or in an incubator.) So it can work well to collect eggs every day, and write the date on them. Then store them somewhere safe (egg cartons or a bowl at room temperature is usually fine.) Eggs more than a week old are still fine to eat, but are somewhat less likely to hatch. Dating each egg will let you tell which ones are worth trying to hatch, and which ones you might as well just eat.
I've probably missed a few details, but I think that is most of what a new chicken owner would need to know about chicken reproduction.