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When using a computer I know when the application is taking the entire screen we call it "full screen" or "your application is in full screen mode".

And so how is it called when a application is taking half of the screen (in order to display 2 applications for instance one on the left and one on the right).

Thanks in advance:

Mostly, it is called "Split-screen mode".

"Half-screen mode" and "Shared-screen mode" works alright too.

In addition to Bella's answer, I occasionally hear the terms "docked" and "snapped" (as in, "The application is docked to the left side of the screen," or "Snap the application to the right side.")

Specifically in the Windows OS, an application window can be in any one of three states; Full Screen, Minimized, and Normal. There is no special term for various arrangements of normal windows on your monitor(s). "Split Screen" is typically used for "couch co-op" games where two (or more) players can each have (all on one monitor) their own dedicated screen space (window) to play the game. "Shared Screen" is typically used to mean sharing the entire screen with a remote third party, like in a Skype meeting you can share your screen with everyone else in the meeting. I've never heard "half-screen" being used, it sounds to me like a property of the monitor itself rather than a window arrangement on the screen.

Edit: Note that in various multi-window applications like Excel, there are names for various arrangements of the application's "inner" windows such as Tiled, Horizontal, Vertical, and Cascade; but those names are more application specific (to Excel in this case) than generally accepted.

I actually create this kind of multi-window arrangement on my screen(s) every day, but I don't have a name or an easy, short description, other than saying something like, "arrange both windows on your screen so you can see as much of each as possible and neither window overlaps" or "imagine drawing a line across the middle of the screen and place one window above the line, one below".

I read in the Collins Dictionary the definition of split-screen and it seems quite accurate, even if it does not explicitly specify half of the screen, the meaning seems close:

[countable noun]

On a computer screen, a split-screen is a display of two different things in separate parts of the screen.

Source COBUILD Advanced English Dictionary. Copyright © HarperCollins Publishers