What part of the plant forms wood in woody stems?

Wood is composed of xylem, vascular tissue that transports water and mineral nutrients from the roots to the rest of the plant. The vascular cambium produces both xylem and phloem.

Wood is formed from the vascular cambium.

The vascular cambium produces more vascular tissue (xylem and phloem), which provide support for the shoot system in addition to transporting water and nutrients. Because the xylem and phloem that come from the vascular cambium replace the original (primary) xylem and phloem, and add to the width of the plant, they are called secondary xylem and secondary phloem. The vascular cambium is only one cell thick and forms a ring around the stem of a plant. On its interior, it adds secondary xylem and on its exterior, it adds secondary phloem. In trees, the layers of secondary xylem form wood. - http://www.shmoop.com/plant-biology/primary-second...

Sclerenchyma and xylem tissue have heavy deposition of lignin in their walls. This forms the wood in woody trees.