Hens produce eggs as a part of their reproductive cycle, just like female humans do, whether a male is involved or not. If you have a rooster, the laid eggs have the potential of hatching into chicks. Otherwise, the hen lays her unfertilized eggs every 24-27 hours (depending upon breed, equating to mostly on a daily basis, though they may miss a day or two a week).
By the way, they also tend to crap a lot and they aren't particular about where they do it. Depending upon the breed, you'll usually end up with 40 to 50 pounds of chicken feces per chicken per year.
In short, if you don't tell your neighbors you have it, they'll probably figure it out fairly quickly from the smell (it might not be bad for just one, but if you have more than one all bets are off).
In case you are wondering, I don't raise chickens. But, about 10 years ago, there was a craze among women about having chicken coops. One of my ex co-workers did it. I still laugh thinking about some of the stories she used to tell about all the "crazy stuff nobody ever told me about chickens". The egg thing was something she found out the hard way - her first batch of chicks got sick and died, so she went overboard on the second batch. They all ended up being hens (except one rooster) - they started laying and she didn't have enough room in her refrigerator for all the eggs she had after a few weeks.
Sadly, she was put out of her chicken misery by a wild mink (think of Hannibal Lecter in the form of a serial killer cat version of a weiner dog) that got into her coop and assassinated all of her chickens...