How do mutations and protein synthesis link to cancer?
Mutation in DNA can be beneficial, harmful, or may have no effect. This is dependent on the gene location where this mutation has occurred.

Various genes linked to cancer are:

Tumor suppressor genes: These genes regulate cell growth by keeping check on cell division, repairing mismatched DNA  and regulating cell death.  Mutation in this type of gene leads to unregulated cell growth and lead to formation of tumor. 

Examples:


TP53 is most commonly mutated tumor suppresor gene. More than 50% of
cancers are caused due to mutation in this gene.
Mutation  in BRCA1 or BRCA2 genes is responsible for breast and
ovarian cancers. It  may also increase the risk of pancreatic cancer
and melanoma.


Proto-oncogenes: These genes help cells to grow normally. Mutation in this gene cause its permanent activation and unregulated cell growth takes place and lead to cancer. This abnormal gene is called oncogene.

Examples:


HER2 is responsible for breast and ovarian cancer.
RAS family of genes, responsible for cell communication, cell growth,
and cell death.


DNA repair genes:  These genes play role in repair of DNA, which is mismatched during replication.   Defect in DNA repair gene, leads to synthesis of incorrectly formed DNA and eventually incorrect protein synthesis.