My mountain bike has 26 x 2.125 tires on it and I am ordering new 26 x 1.95 tires.
Will these be okay to use? What issue might this cause? Or will the difference be negligible?
Yes - you can fit 26x1.95 tires to replace 26x 2.125. They will be slightly narrower, and that might mean you need to pump them up slightly harder. If the tread was the same pattern, they would probably offer slightly less grip on a loose and rough surface, though it is likely that the new rubber will be much more grippy than your old tyre is.