The seed does not develop from just any part of the plant, but from special structures called ovules. The ovule is an immature seed, which does not yet contain a viable embryo. It is only when the egg cell inside the ovule is fertilized by sperm that the ovule is called a seed. The ovule is surrounded by integument tissues which produce the seed coat, and in the earliest seed plants another layer called the cupule enclosed the entire ovule / seed.
Wood is produced by the vascular cambium, a layer of cells whose job it is to divide off cells for new conducting tissues. The vascular cambium is a cylindrical region running through the entire stem of the plant, and branching into every twig and limb. When the cells of this cambium divide, they may produce new cells toward the outside of the cylinder, or toward the inside. Those which split off to the outside of the cylinder become new phloem tissue, which transports the sap of the tree and thus moves food manufactured in the leaves down to the roots. Those cells which split off toward the inside of the cylinder become new xylem tissue, which transports water and minerals up from the roots, and also provides for movement of materials between the exterior and interior of the plant. When a great deal of xylem accumulates, it is called wood; plants with wood may be trees, shrubs, or stout vines.
Periderm is the other product of secondary growth; it is produced by the cork cambium, a cylindrical layer of cells which develops not far under the outer skin (epidermis) of the plant. Like the vascular cambium, the cork cambium divides new cells toward the inside and toward the outside. Those which are produced toward the outside become the particularly important tissue called cork. Cork is important because it replaces the original outer layers of tissue as the plant grows. A young plant begins with a smooth intact surface, but the growth of two cambial layers producing new tissues strains this outer layer, causing it to rupture. Cork is produced to replace these lost tissues, and thus protect the inner tissues. Usually, this new cork will have a different texture, and may even later be replaced by additional new cork. Together, the periderm and the phloem (which lies just to the inside of the periderm) are called the bark.
Secondary growth in stems: Above, you can see sections made from the stem of Tilia (Basswood) made after one, two, and three years of growth. The smallest stem is the youngest, and the largest is oldest.
Secondary growth, as described above, occurs only in seed plants and a few of their extinct close relatives, such as the progymnosperms. Even so, not all seed plants actually produce such secondary growth; the monocots are one such group which lacks this kind of growth. There are a number of other groups which do develop secondary tissues, such as some extinct lycophytes and sphenophytes, and their secondary growth occurs by a similar process, though it differs in a number of important details. The extinct scale trees, for instance, produced woody growth enough to reach heights of several dozen meters, but their primary support was their periderm (bark, not wood), and both cambia were unifacial, that is they divided new cells only toward one side of the cambium, and not to both sides as in seed plants.