The axillary bud is an embryonic shoot which lies at the junction of the stem and petiole of a plant.
As the apical meristem grows and forms leaves, a region of meristematic cells are left behind at the node between the stem and the leaf. These axillary buds are usually dormant, inibited by auxin produced by the apical meristem, which is known as apical dominance. If the apical meristem was removed, or has grown a sufficient distance away from an axillary bud, the axillary bud may become activated (or more appropriately freed from [hormone] inhibition). Like the apical meristem, axillary buds can develop into a stem or flower.
Certain plant diseases - notably phytoplasmas - can cause the proliferation of axillary buds, and cause plants to become bushy in appearance.
Axillary buds can be used to differentiate if the plant is single-leafed or multi-leafed. Simply count the number of leaves after an axillary bud. If there is only one leaf, then the plant is considered single-leafed, vice versa.