If a mutation occurs in the DNA let's say a point mutation only one of the base pairs will be different. The codon will be different so a different amino acid will be synthesized. Amino acids have different radicals with different functional group. Those radicals that differ from amino acid to amino acid determine the shape and function of the protein. Therefore a mutated gene will cause the synthesis of a protein with an altered shape and function. A good example is sickle cell disease. Where glutamic acid is replaced with valine and the beta chain of hemoglobin has a deformed shape and has an altered function(carries less oxygen).
