As in normal usage, the word 'adjacent' in geometry refers to items that are next to each other in a figure. Usually applied to lines, arcs or angles. See below.
In a polygon, adjacent angles are angles that are next to each other. They share a common side. For more on this see Adjacent angles.
Adjacent sides are those the are next to each other. They share a common vertex. For more on this see Adjacent sides.
Trigonometry often deals with right triangles, where the three sides are often referred to as the hypotenuse, adjacent side, and opposite side. Adjacent side is the side next to the angle in question. (The one other than the hypotenuse). For example:
Here, the side A is adjacent to the angle x. For more see Trigonometry functions overview.
Adjacent arcs are those the are next to each other on the circumference of a circle. They can be combined to make a single larger arc. For more on this see Adjacent arcs.