In general 'to bisect' something means to cut it into two equal parts. The 'bisector' is the thing doing the cutting.
In an angle bisector, it is a line passing through the vertex of the angle that cuts it into two equal smaller angles. In the figure above, JK is the bisector. It divides the larger angle ∠LJM into two smaller equal angles ∠LJK and ∠KJM.
The two smaller angles are adjacent angles because they share the common leg JK.
See also Constructions: bisecting an angle to see how to bisect an angle with compass and straightedge.