Interference between two spherical waves 
Two waves with the same frequency and phase (coherent). The circles (wave fronts) represent the points where the wave function takes its minimum value. In between two circles the function takes its maximum value. 
At the points in red there is constructive interference: the two wave functions have the same value (maximum or minimum). The result is a wave with an amplitude twice bigger. At the points in balck there is destructive interference: the two wave funcitons have opposite values giving a null result. The points where there is constructive or destructive interference reamin fixed. At the points of constructive interference, the difference between the distatnces to the two sources is an integer multiple (n) of the wave length. In this particular case n = 9, 8, ..., 0, 1, ..., 9 and to each value of n corresponds one of the red curves. In the points where there is destructive interference, the difference between the distances to the two sources is a halfinteger (m) times the wavelength. In this particular case m = 17/2, 15/2, ..., 1/2, 1/2, ..., 17/2 and to each value of m corresponds one of the black curves. 
Interference between two light beams 
On the screen (green) there are bright points (red) where there is constructive interference. In between two bright zones there is a dark zone with destructive interference.
Since each beam is a cone, the red fringes on the screen are actually concentric circles, as shown on the following image, taken from a real interferometer. The radii of the circles will depend on the wave length and on the distance to the screen. The position of the first circle depends on the ratio between the distance and the wavelength. If the distance between the sources was a integer multiple of the wavelength, the brightest point would be at the center of the interference pattern. If that distance was a halfinteger times the wavelength, the center of the pattern would be completely dark. If the two sources were moving apart from each other, it will appear as if there were circles coming out of the center of the interference pattern. Each circle that goes out would correspond to a displacement of a wavelength. 
Michelson interferometer 
It uses a single light source, and a beam splitter that separates the beam in two, which are sent to two plane mirrors that reflect them to the screen.
If the distances from each mirror to the screen are different, the two waves will seem to come from two points at different distances from the screen. 
The source used is a laser beam, which provides a coherent light. One of the mirrors is fixed while the other can be moved.

The laser used in the interferometer produces a parallel beam. If we consider a single ray of light coming out of the laser, the interference on the screen will be either completely constructive or destructive, depending on the difference of the distances from the two mirrors to the beam splitter.

The actual laser beam is not just a straight line but a cylinder, and the mirrors will have a small deviation from the perpendicular to the beam.
That will imply that the distance travelled by different rays in different parts of the beam will be different, producing zones of destructive and constructive interference in different places of the screen. The interference pattern obtained is composed by parallel dark and bright fringes. 
A displacement of the movable mirror of one fourth the laser's wavelength will increase in one half of the wavelength the distance travelled by one of the beams. At the points where the wave function used to be minimum it will now become maximum; where there was constructive interference, there will now be destructive interference: each bright fringe will move to the place where its neighboring dark fringe was.
That will imply that the distance travelled by different rays in different parts of the beam will be different, producing zones of destructive and constructive interference in different places of the screen. The interference pattern obtained is composed by parallel dark and bright fringes. 