Axillary bud and terminal bud derived from the activity of : -

The shoot apical meristem at the tip of the stem forms the terminal bud and produces new leaves and the primary tissues of the stem.
As the leaves form, some cells of the apical meristem are left behind in the axils of the leaves; these give rise to axillary buds, which can grow into branches or flowers.
Lateral meristems (vascular and cork cambium) produce secondary growth in thickness, and intercalary meristems add length at the nodes of grasses.
So axillary and terminal buds are derived from the activity of the apical meristem.