Growth in plants occurs chiefly at meristems where rapid mitosis provides new cells. As these cells differentiate, they provide new plant tissue.
Fig. 22.214.171.124 Meristem Fig. 126.96.36.199 Horse chestnut
In stems, mitosis in the apical meristem of the shoot apex (also called the terminal bud) produces cells that enable the stem to grow longer and, periodically, cells that will give rise to leaves. The point on the stem where leaves develop is called a node. The region between a pair of adjacent nodes is called the internode. The internodes in the terminal bud are very short so that the developing leaves grow above the apical meristem that produced them and thus protect it. New meristems, the lateral buds, develop at the nodes, each just above the point where a leaf is attached. When the lateral buds develop, they produces new stem tissue, and thus branches are formed.
Under special circumstances (such as changes in photoperiod), the apical meristem is converted into a flower bud. This develops into a flower. The conversion of the apical meristem to a flower bud "uses up" the meristem so that no further growth of the stem can occur at that point. However, lateral buds behind the flower can develop into branches.
The drawing is of a typical woody dicot, the horse chestnut, as seen during the dormant season. The leaves have dropped off, leaving a leaf scar; the dots inside each leaf scar show where the vascular bundles (xylem and phloem) had entered the petiole of the leaf. A flower had been produced the season before, so that during the season just ended two branches had grown out on either side of the flower bud scar. Lenticels are openings that allow oxygen and carbon dioxide to diffuse between the living cells of the stem and the air.
The growth of roots in described in a separate page — Link to it.