The egg laying cycle of a chicken goes something like this:
0-6 months: no eggs
6 months- 2 or 3 years- 3-6 eggs per week
Molting- 4-6 weeks during the fall (usually October-November)
Winter- once the days get shorter, usually they don't lay again after molting until February. The only exception is with chickens that have just started laying- they will usually lay through the winter months during that first year.
Broody- depends on the chicken. Some will never be broody, some are broody constantly. Basically they sit in an empty nest box thinking they are hatching eggs...
For the sake of conversation, let's say that you have a flock of 10 backyard laying hens.
Buy five chicks in April of the same breed (actually probably get 6-7 because they hardly ever all make it to adulthood). These chicks will begin laying in October, lay through the winter, really ramp up egg production in the spring and then take their first rest in the fall of the second year.
Buy five chicks in April of a second breed, preferably that lay a different color egg than the first. These chicks will begin laying in October of the second year, just as the first group begins to take their rest for the winter.
Buy five chicks in April of either a third breed or the same as the first breed, once again that preferably lay yet a different color egg. These chicks will begin to lay in October as the first group finishes their second year of laying and the second group takes their first rest. As this group of chicks begins to lay in October, it will mark two years of laying for the first group, which means their egg production will have significantly slowed down. The first group of chickens will be retired (to the pot or freezer if you want to make the most of your investment).
Year 4 and forward:
Continue the cycle of buying new chicks each spring and retiring the oldest group of layers each fall to keep up optimum egg production in older hens, year-round egg production from the youngest hens, and a little meat every fall for chicken noodle soup or gumbo!
Many people start a flock of backyard chickens without considering that the group will go through its best production, as well as breaks, and finally slow down and stop laying all at the same time. Staggering the ages of your chickens will help even out the overall production of the flock over time. Also, using groups of chickens with different colored eggs will help you keep track of the production of each individual age group of chickens within your flock. I hope this helps you plan your long-term chicken flock!