The phenomenon of Newton's rings is an interference pattern caused by the reflection of light between two surfaces - a spherical surface and an adjacent flat surface.
The brightness or darkness depends on the height of air column between the spherical surface and the adjacent flat surface.
The locus of all points on the flat surface which are having the same height of air column is a circle whose center is the center of contact of the two surfaces.
Hence light reflected from all these points will have the same brightness. Thus it forms a ring of equal brightness. The points where there destructive interference takes place are lying in a circle and hence dark ring is formed.