Explain to a layperson how cigarettes smoke might cause cancer
I encountered the following paragraph in the textbook Life: The Science of Biology:


  Some chemicals add groups to the bases. For instance, benzopyrene, a component of cigarette smoke, adds a
  large chemical group to guanine, making it unavailable for base
  pairing. When DNA polymerase reaches such a modified guanine, it
  inserts any one of the four bases at random. Three-fourths of the time
  the inserted base is not cytosine, and a mutation results.


After finding the following quote in Wikipedia's Benzo(a)pyrene page, I decided to explain specifically how BaP can cause cancer:


  BaP has since been identified as a prime carcinogen in cigarette
  smoke.




My explanation:


A protein molecule is a chain of amino acids. Proteins are hugely important, as they are responsible for most of the cellular functions.
A DNA molecule is a chain of nucleic acids. There are 4 types of nucleic acids in DNA: A,G,C,T.
Having intact DNA molecules is crucial, because they include the encoding of proteins, according to which cells synthesize proteins.
A mutation is an alteration of a DNA molecule.
A mutation might lead to our cells synthesizing a malfunctioning protein. If a protein that should regulate cell division is malfunctioning, then the result might be uncontrolled cell division - cancer.
When you smoke, the chemical BaP (which is found in cigarettes smoke) might alter a G nucleic acid in a DNA molecule in some cell in your body. Later, when this cell tries to divide, it makes a copy of each of its DNA molecules, but it won't be able to identify that G nucleic acid (that BaP altered), so it would put a random nucleic acid in the copied DNA molecule. Thus, there is a 3/4 chance for a mutation.