Carpenters produce items out of wood. On wood-scarce maps, their main use is to build beds, as every other essential item in the game can be made from other materials. However, on Forest maps or maps where wood is more common, they can be used to produce essential bins and barrels, and all manner of other furniture except statues. They can also create structures such as walls and staircases, which is very useful on maps where most stone is locked away under an aquifer.
To employ a carpenter, create a task at a carpenter's workshop or designate a construction.
Some players ignore carpentry, since all wood only has a material multiplier of 1 (the lowest possible), and the value will never amount to much. But wealth is only one consideration - all beds are built by carpenters, and all dwarves need a bed - and a high-quality bed can give happy thoughts, which become more valuable than gold sooner or later. That's a guaranteed happy thought to every dwarf every day.
It is notable that shields are both lighter weight than furniture, and also worth half again as much (25r weight and 15☼ base value for shields). This can come in handy if, for instance, you have a legendary carpenter but no skilled masons or craftsdwarves and are desperate for trading items.