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.
Opie's Ill-Gotten Gain - S4-E8
Continuity mistake: When Opie's surprised by the bike, his first closeup is a flipped shot - note the bell's on the wrong side of the handlebar, etc.
Visible crew/equipment: When Opie's up on the ladder feeding the birds, in the overhead shot of Andy and Barney the mic's cable running up under Barney's pant leg is visible.
Aunt Bee the Crusader - S4-E15
Revealing mistake: Toward the end of the show the rooster, Bo, is drunk. If you look closely you can see the two strings pulling the rooster's neck sideways.
A Black Day for Mayberry - S4-E7
Continuity mistake: At the beginning of the show, two men walk into the courthouse. (As a sidenote, one of the men is Rance Howard, which is Ronnie Howard's real dad). Barney is the only one there, as Andy is out. The two men sit down to wait for Andy. Rance Howard is not wearing a hat, and he is in a light-colored suit. When Opie walks through the door looking for Andy, you can see that Rance is not sitting there any more, and now it's a man with a dark suit and wearing a hat. (00:01:55)
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.
Visible crew/equipment: While Aunt Bee, Andy, Barney, and Gomer are on the porch talking about the band, just as Andy says, "We're talking about gettin' up a band concert by tonight," the chalk T-mark is visible on the floor by Andy's feet. (00:12:25)
Ernest T. Bass Joins the Army - S4-E3
Revealing mistake: When Barney picks Andy up and they get back to the courthouse, they find Ernest T out of the jail cell again and breaking more windows, and in the shot of Ernest T leaping in the air we see it's actually a flipped shot - note the backwards "snack bar" in the window (that store is later when Ernest T tries to break the new glass for the courthouse).
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.