A good example of this is actually human eye colour, although I am going to be simplistic here. You probably know that brown eyes are dominant over blue eyes, and can probably guess by now that the "recessive" gene that causes blue eyes is actually a lack of brown-producing gene. The fun bit is that blue eyes are not actually a *complete* lack of pigment. People with blue eyes still produce a small amount of the brown pigment melanin, which over the unpigmented pinkish colour of the iris, produces a bluish colour. It's worth noting here that albinism is the "no pigment at all" version of this mutation This is, again, a simplistic view, as there are actually several proteins which are necessary for melanin production, but we can ignore that for the moment.
So, how does that work? We'll take a hypothetical pigment producing protein here, which picks up a mutation that converts a positively charged amino acid to an uncharged one of about the same size. The protein isn't entirely happy; it's not holding together as well as the unmutated (wild type, to use the technical term) version. However, it just about clings on, and manages to catalyse the reaction to make melanin at maybe 20% of the rate the wild type would. It's just not enough to make brown eyes, but it is enough to make a difference to the colour of the iris.
And yes, there are mutations that don't really have an effect on protein function at all- sometimes you can insert, delete or change an amino acid and be entirely unaware of it. It's also possible to have one mutation cancel out another- this is termed a "rescue" mutation in genetics. To drag out our example of the positive-to-negative-charge mutation again, if our hypothetical protein with a positive instead of a negative charge at the crucial place on the binding site meets its binding partner that has a negative instead of a positive, the interaction may well work, when if only one of the two mutations were present, it would not.