How does a gene mutation alter a protein?

A genetic mutation, defined as being a change in the DNA base sequence would lead to a change in the primary structure of a protein (a change in the amino acid sequence). This would lead to a change in the secondary structure of a protein as hydrogen/ionic bonding within the B-pleated sheet or the A-helix would change. This would therefore lead to a change in the final 3D folding shape of a protein (the tertiary structure) as hydrogen/ionic/hydrophilic/hydrophobic/disulfide bridge interactions would be affected, which would alter the function of a particular protein.
The way in which it alters the function is entirely dependant on what mutation occurred. Some mutations will have no effect (silent). They could boost the function of a protein (eg active site of enzyme more complementary to it's specific substrate) or they could reduce the function of a protein (eg active site of enzyme no longer complementary to its specific substrate).