The Forest in OpenTTD is a plantation of pine trees used for making either timber products or paper (in the Temperate and Sub-Arctic climates respectively). The trees are automatically replanted by the forestry staff ensuring a continuous supply of Wood for you to transport.
In the Sub-Arctic climate, forests will only exist above the snowline. If you are having problems generating maps with forests, make the maps more hilly or even mountainous from the map generation screen.
Forests are not available in the Climates climate since pine trees are not suited to it. Instead, sub-tropical has the Lumber Mill.