The woodcutter worker in the woodcutter building uses logs to make firewood. Firewood is governed by the fuel limit. This starts at 200. On a medium start, we already have 200 firewood in the stockpile so no more will be produced until somebody takes some home for a private store ready to heat their house. This will not happen until we complete a house.

In the basic game, woodcutters compete for logs with builders, who use them directly. Logs come from cutting down trees. Foresters do this only if the reserve of logs in the stockpile is less than the log limit (which is the first one on the list). Laborers told to harvest trees (or who have to do so incidentally in preparing a building site) produce logs regardless of the limit.

PS:
After completing a woodcutter building, remember to assign somebody to the woodcutter job! (To be pedantic, tell the work manager that one person is needed - we have no direct control over who it will be.) There is a different warning symbol for a workplace without a worker, and another for one lacking its raw materials.

Oh, and when the WORK button on the display we get by clicking on a building is lit up, this means the building IS meant to be working (everything else permitting) so clicking it means "DO NOT WORK". When the button is faded out, that is when clicking it means to start work.