The sawmill is the resource producing building that produces the lumber you need.
Lumber tends to be the most expensive resource, so many players build lots of lumber cities that are only focusing on producing lumber. They will tell sell extra lumber that they have and buy the other resources that they need.
You will use lumber both for building construction as well as troop building, that’s why it is so important. Almost all troops you can build in Rock Age Online need lumber to be trained. For many troops lumber is also the most important resource when it comes to training them. You should build lots of sawmills in each of your cities for this reason.