Q: What is a nest box?
A: A nest box or nesting box is just an enclosed area meant to give chickens a place they feel comfortable laying their eggs.
Chickens generally prefer darkened, enclosed places where they feel safe. We say "generally," here, because nee layers may not understand what their instincts are telling them to do. Many lay an egg or two on the floor. Some feel like they need to find a spot outside the coop to lay (such as under your porch!), and so on. It may take some time and effort to "teach" them to lay in the coop. And some may always prefer to lay in a corner than in a nest box! Read the related questions below for more information about that.
The next boxes themselves don't have to have any particular construction. Some may be plastic, some metal, some wood.
Plastic next box
Some will be purpose built, and others will be improvised. For instance, many people will set 5 gallon buckets on their sides, with bricks on each side to keep them from rolling, and hens will lay in that. One member of the MPC team found her silkie bantams liked to lay in tangerine crates set on the coop floor. There are "roll away" next boxes that allow the freshly laid eggs to roll out of the way once laid (preventing accidental breakage).
Some coops have built-in nest boxes (shown above) arranged so that egg collection can occur without having to go into the coop.