DNA codes for proteins. So, if the DNA is mutated, it directly affects the protein. Two examples of mutations are:
A point mutation. In this situation, one base pair (A, T, C, or G) is replaced with a different, incorrect one. When this happens and the DNA is read in order to synthesize a protein, this incorrect base pair can lead to the DNA codon (set of 3 base pairs) that codes for a different base pair than it should have. This causes the protein to have one incorrect amino acid in it. Also, this mutation does not always mutate the protein because some codons code for the same amino acid.
A frame shift mutation. This type is much worse. In this case, a base pair is just inserted in to the DNA sequence. When this happens, the entire sequence following it is essentially pushed over one spot. Then, each set of 3 base pairs that would code for an amino acid is shifted so that, instead, you would have something like 2 base pairs and 1 base pair from the codon next to it. This causes every single codon to mutate, which essentially leads to every amino acid being different and therefore, the entire protein being different.
I hope that made sense. Diagrams help.