When I say "verticality" I'm referring not just to the outside shape of the canvas of paper or whatever the picture is on, but also to dominant lines and directions within the picture.
Recently I've written some posts about horizontality in art in which you can read about (and see demonstrated) many different effects you can achieve through the use of horizontality. As you can imagine, to achieve opposite effects you might be able to use verticality instead (though it's not always quite that simple).
For example, if your goal is to emphasize a relationship between equals you might want to use a horizontal format, with whatever you want to portray as "equal" on the same horizontal level, but if you want to show who's (or what's) in charge, you might want to use a vertical format, with the one who clearly has the advantage in the situation above the one who is at a disadvantage, for example in the following picture.
Single Combat of Prince Mstislav Udadloi with Rededia - 1812
Andrei Ivanov, Russian painter - 1775-1848
In the above picture it's easy to see who is in control. (The angel above is "floating" and obviously not affected by gravity which is what (perceptually) determines who has the advantage; what is important is the relationship to the ground of whatever appears that it would be affected by gravity.) Of the two men in the fight (it was actually considered a "duel") we can probably safely assume that the eventual "winner" (Prince Mstislav) is the man in a vertical position (slightly inclined toward the other man in the competition in a threatening way) and who also is "above" the obvious loser who is apparently about to become permanently horizontal. The verticality of the outer shape of the picture helps to emphasize the dominating-dominated relationship shown by the vertical hierarchy within it.
The Card Players - 1890-92
Paul Cezanne, French painter
Source: The Athenaeum
In the above picture none of the card players is seen as having an advantage. They are all at the same level horizontally. Although the man in the blue smock against the wall is "above" the card players he is smaller and obviously at a distance behind them, has his arms folded as if holding himself back, and seems busy smoking his pipe, so he obviously is neither in charge of nor threatening to the card players. Often a servant or a bystander is shown standing behind a main character in a picture but they are not seen as in the space "above" that main character ... they are obviously at a distance and are played down in other ways also ... such as with subdued colors, lack of contrast with the background, etc.
There are other posts on this blog that have to do with horizontality and verticality. Look at the right side of the page for keywords to get to a page with all the posts that have been tagged by me with one or both of these keywords. There are also other posts that you might be interested in that may not be on that page -- Most anything to do with the Cartesian grid, for instance. There is a search box for this blog on the right side of the page, also, though I admit it's hard to find because the list of keywords is so long.
"When adversaries meet on a horizontal base they are equally matched by their spatial position; in a vertical composition, however, having the upper hand spatially constitutes an advantage, whereas being in the lower position means having to overcome the pull of gravity as well as the opponent's onslaught." (Rudolf Arnheim in The Power of the Center, page 96)