Definition: Two angles
that share a common side and a common
vertex,
but do not overlap

Try this Drag the orange dot. The line AC is the common leg of the two adjacent angles

In the figure above, the two
angles∠BAC and
∠CAD
share a common side (the blue line

AC

).
They also share a common
vertex
(the point A). They are therefore termed 'adjacent angles'.

Obviously, the larger angle ∠BAD is the sum of the two adjacent angles.

They do not overlap

In the figure on the right, the two angles
∠PSQ and
∠PSR overlap.
Although they share a common side (PS) and a common vertex (S), they are not considered adjacent angles
when they overlap like this.
Adjacent angles must be next to each other, not one on top of the other.

Another way of defining them is:
"two angles that share a side and a vertex, but do not share any
interior points".

In Polygons

Another use of the term refers to the interior angles of polygons.
Any two interior angles that share a common side are called the "adjacent interior angles" of the polygon, or just "adjacent angles".
Here the word adjacent is used in its ordinary English meaning of "next to each other".