Deliberate mistake: During scenes at Walker's Drugstore there are many magazines by the wall near the door, which have tape/paint hiding part or all of their cover titles. Examples are: all the Sports Illustrated have tape across the word "Illustrated", the House and Garden has its "House &" covered, and Time magazine's entire title is blacked out.
Continuity mistake: At the "Hen Party", Jeff Pruitt says hello to Thelma Lou and she states she is going to get coffee, then goes to the kitchen. However, when Jeff walks across the room and looks back, Thelma Lou is back in the same area. She never walked back from the kitchen.
Factual error: When you walk into the courthouse the first jailhouse to the left has a window. But the courthouse is connected to another establishment on that side. A window would be impossible.
Continuity mistake: In the initial episode, and Runaway Kid in season 1, Barney is a cousin of Andy's. In subsequent episodes, they are unrelated.
Factual error: The badges they wear don't match the badges on their patches. If there's a badge on the patch, it's supposed to match their actual badge.
Other mistake: In some episodes Barney can sing beautifully, but in others he can't sing at all.
Continuity mistake: Scenes shot at Andy's porch have significant differences depending on the filming location, as there are two sets of Andy's porch. One is just the facade of the house in the 'town of Mayberry' in the studio's 40 Acre backlot; and the second is located in the studio's soundstage. Note in some shots the door's sidelights have vertical muntin bars, the screen door has a white handle, whereas in other shots the sidelights' muntin bars are gone and the screen door handle is black (screen door's gone season 5). Also, note the significant difference in the width of the wall between the parlor windows and the door, the height of the parlor windows from the floor, etc.
Other mistake: The radio communications consoles on the side desk in the courthouse have a few pieces of black tape covering their brand name (though we know it's Motorola from the visible "M" logo).
Revealing mistake: On most of the corners of the buildings, you can see seam lines running down between the different sides of the buildings. The bricks on the opposite sides usually don't line up with each other either even though they are supposed to be the same brick.
Continuity mistake: The fuel pumps at Wally's gas station (regular and super) switch sides between numerous episodes.
Factual error: A town like Mayberry with a mayor and council would have its own police force, not just a sheriff.
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.