Continuity mistake: At the end, when Andy walks in on Barney showing Gomer how to draw his gun, Gomer draws his gun out of the holster which sends it flying behind him, and in the closeup the bookcase glass shatters, but in the wideshot when they all turn to look the glass is still perfectly intact.
Continuity mistake: When Gomer drops the gun from the roof it lands on the ground directly in front of Andy and Barney, who are standing on the sidewalk beside the courthouse, but the area where they're standing looks different between the overhead shots and closeups, and their shadows change as well. Additionally, the gun lands with its handle to the left, but in the wideshot the gun is lying the opposite way, with its handle to the right.
Opie's Group - S8-E9
Continuity mistake: While the Taylors are having dinner and talking about Opie's band, Opie has a filled dinner plate, salad plate, glass of milk, and flatware spread out on the table in front of him, but it all disappears in the shots facing Aunt Bee and Andy.
Continuity mistake: After Howard's appearance on the TV show, when he and Andy get back to Mayberry and talk to Opie on the porch, both Andy's tie and Howard's bow tie have striped designs that are positioned differently once they walk into the house, which could only happen if they were retied. (00:15:00)
Continuity mistake: At the start, when Andy stops at Wally's station Gomer is telling Jason about his Trout bait, then someone pulls up behind Jason's car and they're bumper to bumper in the wideshot, but in the exterior shot facing Jason there's no car behind him, but that car is back again before he pulls away.
Continuity mistake: When Harry hangs the sign on Floyd's shop it reads "For sale Harry Walker real estate," but in the closeup of the sign the word "by" suddenly appears and it reads "For sale by Harry Walker real estate", then it disappears. It looks as if it reappears in the wideshot when Howard stops to read the sign later. (00:01:15)
A Visit to Barney Fife - S7-E18
Continuity mistake: While Andy and Barney are sitting on the porch, Barney's clip-on striped bow tie is clipped on the opposite way (stripes flip) when Ma and her family get back from "shopping." (00:14:45)
Continuity mistake: When Andy and Floyd stop by Wally's station, Flora's wearing a striped uniform shirt and there are a bunch of cars lined up waiting for service. Then later, when Goober gets back from vacation and shows up at the station, in the first shot of the station Flora's wearing the striped shirt and all the cars waiting for service are the same as the earlier shot, but in the closeups Flora's now wearing a solid color uniform shirt. (00:06:05 - 00:09:20)
The Gypsies - S6-E23
Continuity mistake: While Andy and the townsfolk are at the courthouse during the storm, when The Gypsies walk in the two men are wearing wet rain ponchos and the woman is wearing a striped serape, but then she's wearing a wet raincoat, and it switches yet again in following shots. (00:21:45)
The Legend of Barney Fife - S6-E18
Continuity mistake: At the start of the episode, while Barney's in Andy's kitchen he's wearing a shirt with large distinctive designs, but when he and Andy walk into the courthouse Barney's shirt has completely different designs (though it's the same type of shirt). (00:01:00 - 00:04:00)
Aunt Bee Learns to Drive - S6-E21
Continuity mistake: The car which Aunt Bee buys from Goober has a rearview mirror which repeatedly disappears and reappears between shots, throughout the episode.
Wyatt Earp Rides Again - S6-E20
Continuity mistake: After Opie tells Andy that he knows he'll win at high noon, it cuts to Warren in front of Floyd's shop (with its furniture conspicuously missing), but later when Andy walks over to Floyd's shop its facade has changed. Note differences such as window structure, and the name "Floyd's Barber Shop" painted on the window which wasn't in the earlier shot. (00:18:05 - 00:24:15)
Continuity mistake: As Barney is turning in his deputy equipment to resign his handcuffs are hanging out of his pocket. After Barney states he wishes someone would just commit a crime, Opie enters. As the camera pans out so Andy can ask Opie what happened, the handcuffs are no longer in Barney's pocket, and he didn't put them on the desk with the rest of the equipment. (03:03:00 - 03:11:00)
Continuity mistake: When Andy finds the old newspaper clipping in Helen's box he tucks it into his pocket, but when he's looking at the clipping while he's at the office the newsprint on the back is entirely different. Then when Howard reads it the newsprint changes yet again.
Aunt Bee's Big Moment - S8-E23
Continuity mistake: When Opie asks his father how to spell the word "Renaissance," Aunt Bee holds a plate of assorted cookies, but when she stands up and heads to the kitchen the type of cookies on her plate have changed.
The Pageant - S5-E11
Continuity mistake: While Andy and Opie are sitting on the couch, Andy's on the phone talking about having to tell Aunt Bee to drop out of the play, and there are two candlesticks with candles on the server behind Andy, but in Andy's closeups they're gone.
The Arrest of the Fun Girls - S5-E28
Continuity mistake: When Helen and Thelma Lou walk into the courthouse and find Andy and Barney dancing with Daphne and Skippy, there are lipstick marks on Barney's cheek, but when Andy gives Skippy her sweater and radio then tells them both to go, in the closeups the lipstick marks are gone, though they reappear momentarily.
Opie Flunks Arithmetic - S5-E30
Continuity mistake: At the end, when Helen shows up at the courthouse and tells Andy that Opie got a B+ on the test, her purse vanishes and then reappears while she's sitting on the banister beside the desk.
Opie Flunks Arithmetic - S5-E30
Continuity mistake: When Barney, Andy, and Aunt Bee are at the table after Opie goes upstairs to study, in shots that face all three of them there are things on the middle shelf to the left and right of the center plate, but then in other shots either the left item is gone, or the right item is gone.
Continuity mistake: Barney tries showing Andy what the "part bloodhound" dog can do for The Manhunt so Barney blows a dog whistle, and in the closeups of the growling dog the rope that was tied to his collar is suddenly lying on the floor, but it's back on his collar in the following shot.
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.