Trees are the source of Logs in the game which also makes them the key ingredient to fire. While other things can be used to make or feed a fire, nothing does so like wood does. A single log can grow a fire while it may take several of other fuels to do the same.
There are six forms of trees: Baby, Small, Medium, Large, Dead, and burned. After you plant a pine cone, you'll immediately see a Baby tree which can't be harvested for anything. Dead trees are what happens after a tree has run the course of it's life and takes the form of just a withered tree with no leaves. If you cut one of these down, you'll be rewarded with a single pine cone allowing you to replant the tree. Burned is the state of a tree that has caught fire and gone out. Cutting this tree down will produce a single piece of Charcoal.
The rest of the stages of the Tree can be cut down for wood and will generally be pretty prevelent. Trees will grow from small to medium to large over time. The amount of resources you get from a Tree depend on the size and it is to your benefit to chop only Large trees if possible. Small trees will only give you a single Log, medium trees two Logs and a pine cone while a large tree will give you three logs and two pine cones. If you let the tree stay large for too long however, it will die and you will only get a single pine cone from it (essentially a waste of the entire trees growth time).
If you have a Shovel then you can dig up a tree stump for another Log. This is particularly important if you are growing your own tree farm because you won't be able to replant trees in the same location until you remove the stumps of the older trees first.