The Cast Showoff: Griffith was able to showcase some of his singing skills in a few episodes.
Characterization Marches On: He and Barney are cousins in the first two episodes, something that isn't brought up again; also, for much of the first season, Griffith played Andy as more of a country bumpkin than the straight man role he played in later episodes.
Flanderization: Andy being the Only Sane Man in Mayberry becomes more and more obvious, especially in the last three seasons, almost to the point where it seems he harbors some contempt for the Eccentric Townsfolk around him.
This was originally planned to be reversed, with Barney being the Straight Man to Andy's Wise Guy. However, the creators (and Griffith himself) was quick to realize that the show was funnier with Andy as the Straight Man.
Technical Pacifist: Hates carrying guns unless the situation calls for as such and he needs to borrow Barney's (one-bullet) gun, preferring instead to outwit criminals rather than rough them up.
What Could Have Been: Originally, Andy was supposed to be the comedic lead with Knotts as his straight man (similar to their roles in No Time For Sergeants, but they quickly discovered it was funnier the other way around: "By the second episode, I knew that Don should be funny, and I should play straight."
Expy: While Knotts carried his comic persona to most of his roles in general, Ralph Furley of Three's Company was more or less an identical character in personality, albeit with a flashy style of dress and different demeanor towards women.
Non-Action Guy: In WWII, at least, in contrast to Andy. Barney was never allowed to leave his post in office as a file clerk, while Andy was stationed in France.
Official Couple: With Thelma Lou at first, but the two call it quits when he leaves Mayberry. In Season 6's "The Return of Barney Fife", Barney discovers Thelma Lou has married someone else. But, in 1986's TV movie "Return to Mayberry", the couple end up getting officially married.
Small Name, Big Ego: Ignoring all that happens under his nose, Barney considers himself an expert lawman, as well as a specialist on women, singing, history, etc. Andy regularly lets Barney carry on as he pleases, although he only allows him a single bullet in his gun for prime safety reasons.
All Women Want Bad Boys: In the early seasons, Aunt Bee has romances with several men that turn out to be complete jerks. Averted in the color era, when she finds happiness with respectable gentlemen such as a professor and a congressman.
The Bus Came Back: In-universe, when Aunt Bee returns to Mayberry in the show's first episode after living in West Virginia for 5 years.
Parental Substitute: For Opie's mother. In the first episode, Opie was originally not too happy with accepting another mother figure into the household, but he warmed up to Aunt Bee by the end of the episode and they became instant friends.
Reality Subtext: The show's presence in Mayberry endeared Bavier to move to adjacent Siler City in real life.
The Pete Best: McNear took over the role of Floyd after Baldwin's one-time appearance as the character.
Real Life Writes the Plot: McNear died less than a year after the final episode of Season 7, so that he was not able to return for Season 8. A Suspiciously Similar Substitute named Emmett Clark, a fix-it man, was added to fill in the character's absence. In-universe, Floyd was announced as having retired because he had earned enough money.
Reality Subtext: As the result of a stroke, McNear was always shown from that point on sitting down or leaning against an object because the actor could not walk. Declining health caused him to leave the show entirely (see above).
Official Couple: After Barney leaves Mayberry, she ends up marrying somebody else, and the episode "The Return of Barney Fife" in Season 6 serves as her last appearance on the show. However, she does return for the 1986 TV movie Return to Mayberry, where she and Barney end up getting married.
You're Just Jealous: In the episode where Andy and Barney help a friend of theirs find a bride and he selects Thelma Lou as the girl he wants to marry, Thelma Lou claims this is the reason Barney is so jealous, although she never wanted to marry anyone else in the first place, and she and Barney get back together at the end of the episode.
Catch Phrase: Not catch phrase, per se, but he did have a habit of asking, "Huh?" after a long succession of questions.
Chuck Cunningham Syndrome: He disappeared without a trace after eleven appearances in Season 6. In the 1986 TV movie "Return to Mayberry", it is mentioned that he became Sheriff of Mayberry after Andy left, and remained at that post until his death.