The main effect of placing a refractive medium in one of the beam paths is that the phase of that beam will be delayed by an amount depending on the refractive index of the medium and the thickness of the medium. If the phase of one of the beams is delayed slightly, the interference fringes will shift laterally. Delay the phase by 1/2 cycle, and the fringes will shift by 1/2 of their spacing.
Note that the whole fringe pattern will not be shifted laterally. The fringe spacing (which is greatest in the middle and gets smaller toward the outside) will not be changed in the pattern; only the locations of the bright and dark fringes will change. In other words, for small shifts the amount of lateral shift is proportional to the fringe spacing. If the amount of shift is an integral number of full cycles, then the fringe pattern will return to its original form. That's not exactly true. If the total phase shift is greater than the coherence length (in cycles) of the light source, then the light from the two slits will not interfere to form fringes. Fringe contrast is a function of the mutual coherence of the two beams.
So, a two-slit interferometer with an adjustable-thickness refractive medium in one path can be used to measure the coherence function of a light source.