The woodcutting center is a production building that harvests accessible forests around it using it's workers.
Forests and woodcutting centers have a very close connection similar to that of rocks and stone mines. The info panel of a forest has serval options relating to the woodcutting center: cut down and let it regrow, protect and do not cut, down and uproot, these will change the way that woodcutters interact with them. You can also prioritize a tree by clicking it and clicking the prioritize button, which will focus all woodcutters with accessible means on the prioritized forest.