"the son of Mr. and Mrs. Adams" means that there's only one son. "The" is a definite article. It's used for a specific thing that the people in a conversation already know about.
"a son of Mr. and Mrs. Adams" means that they have more than one son. "A" is an indefinite article. It's used when people in a conversation don't already know what thing (in this case, what son) you mean.