Continuity mistake: When Opie surprises Andy by telling him that he wants to take piano lessons with Clara, he heads to the piano to play what he already knows, and when Opie plays Chopsticks on the piano he's wearing a different shirt.
Revealing mistake: While Andy, Aunt Bee, and Opie are at the Hollywood studio watching a scene being filmed, when Andy mistakenly stands up and interrupts the filming we can see the wall and shelves behind the director, and it should look familiar because it's the wall and shelves from Floyd's shop in Mayberry.
Visible crew/equipment: While Andy and Otis are talking about women Barney opens the door and rushes into the courthouse, and behind him a crewmember's hand (looks like a woman wearing bracelets) can be seen catching the door as Barney swings it shut.
Continuity mistake: After Frankie pays for the chewing tobacco in the drugstore, in the next closeup of Ellie the cigar box on the shelf behind her has black lines scribbled over all its identifying brand names (which is Van Dyck 43's), but when Frankie walks out the door, in Ellie's next semi-closeup that cigar box is unmarked again.
Visible crew/equipment: When the squad's siren gets stuck blaring, a crowd gathers around the squad with Andy, Barney and Rogers, and when the crowd disperses we can see at least three chalk T-marks on the ground.
Continuity mistake: When Andy and Barney catch up with the man who ran Fletch's truck off the road, the rearview mirror suddenly vanishes in the closeups that face the windshield on Bailey's car.
Andy and Opie, Bachelors - S3-E4
Revealing mistake: After Andy and Opie choose between chicken and pot roast using a nickle, when Opie gives the nickle back to Andy the window over the sink is missing - the fake window wasn't put up on the set wall for this shot, but when the food burns the window and curtains are up there as usual.
Visible crew/equipment: At the Scobey house, after Ben sits in the unstable chair and it breaks, when Andy and the Scobeys rush over to him we can see the two curved chalk actor's marks on the floor, where Ben had been standing moments before.
Continuity mistake: While Andy and Barney are trying to sell The Cannon, when they're talking to the shop owner who's seated in the rocking chair we can see the area between the squad car and storefront, with various items for sale, but in the next shot the covered cannon has suddenly appeared - and it's tall enough to have been in view in the previous shot had it actually have been there.
Visible crew/equipment: When the new manicurist tells Floyd that she works only on commission, she lifts the small table and moves it to the window, and we can see the actor's tape mark where she had just been standing.
Aunt Bee's Big Moment - S8-E23
Visible crew/equipment: After Aunt Bee's solo flight, just as the flight instructor walks over and gives Bee the papers as proof that she soloed, the large T-mark (rubber/plastic) can be seen on the ground between Bee and the instructor.
A Plaque for Mayberry - S1-E25
Visible crew/equipment: While everyone's waiting for Otis to arrive at the plaque presentation in Mayor Pike's office, after Andy admits to the mayor that he asked the real Otis to show up, it cuts to the mayor walking behind his desk, and we can see the chalk T-mark on the floor where Mrs. Bixby stands.
Runaway Kid - S1-E4
Visible crew/equipment: After Andy talks to George's father on the phone, he gets up and stands near the fireplace while explaining to Opie why he broke his promise and called George's father, and when Opie's on the couch asking if he can run off sometime, we can see the white tape mark on the rug where Andy was standing moments ago.
Aunt Bee's Invisible Beau - S5-E27
Revealing mistake: When Clara and Aunt Bee are talking in the kitchen, in the closeups facing Aunt Bee we see the counter, the cabinet, and the wall, but what we don't see are the curtains, the window, or the windowsill, which all should be on that wall. The fake window wasn't placed on the set wall for Bee's closeup.
Continuity mistake: While Aunt Bee and the party guests are gathered in the parlor, when Gomer hoots like an owl in the shot facing him note Floyd's eyeglasses - black with thick straight temples, but in the shot from behind Gomer and all following shots Floyd's wearing different eyeglasses - lighter color with curved tips.
Family Visit - S5-E3
Visible crew/equipment: While Ollie's making up excuses to rush home, he and Nora are saying goodbye to Aunt Bee, and the reflection of the moving boom pole and mic are visible on the car's windshield.
Visible crew/equipment: At the start of the last scene, when Andy and Rogers walk out of the courthouse the reflection of a crew member is visible on the surface of the car, and as Rogers drives away we can see the reflection of the camera dolly on the car, and also the shadow of the boom mic on the ground.
Aunt Bee's Big Moment - S8-E23
Visible crew/equipment: After Aunt Bee tells Opie that she will solo the next day, when she sits in the chair to practice her flight maneuvers, the actor's tape mark can be seen on the floor beside her feet.
Goodbye, Sheriff Taylor - S5-E10
Continuity mistake: While Barney's trying to instruct the new deputy candidates, when he throws a tomato to Otis it splatters on his shirt, and a few things keep changing depending on the angle of the shot - the amount of tomato goo increases/changes position; the suspender's sliding adjuster changes height; the deputy badge changes position.
Revealing mistake: At the end, in the shots of the kids dancing in class, the width of the wall between the shelf/coat hooks and the perpendicular wall with the door has bizarrely increased by 1-2 feet, but that extra width is gone in previous and following shots. Presumably they needed extra space for all the kids to dance so they moved that left wall.
Answer: Those "boxes" are valences that used to be very common, before air conditioning. They allow for windows to be open during rain storms. They permit air circulation, without letting the rain in.