What is landscape mode in photography?

In simple terms, television is landscape and phone screen is portrait.

Rectangle with long edge horizontal - opposite of 'portrait' where long edges are vertical.

6 x4 , shaped like a letter envelope , narrow edges are vertical .

Landscape is wider than tall. Portrait is taller than wide.

Building upon Sumi's answer, I would add that subtle changes in white balance and color saturation are programmed into landscape mode as well as other modes.

I assume you refer to 'Modes' within the camera. These days cameras offer many Modes and Scenes, each of which is intended to give the best rendering for particular types of scene you are photographing. So Landscape is best for photographing  countryside and canyons, etc.

But it occurs to me that possibly the "Landscape" you are being asked about is simply the fact of holding the camera sideways. The opposite is "Portrait", holding the camera upright.

To understand any of these picture-specific modes, one needs to understand what the camera does when placed in the traditional P or program mode or the green auto position.

When the camera takes a meter reading to determine the necessary aperture and shutter speed, it looks at the light reflecting off of the scene through the lens.  The camera has no way of knowing what it is photographing. S how does it know what aperture and shutter speed to use?  The camera uses an algorithm where it will chose a combination of these settings based upon the brightness level of the scene.  Often the camera will favor a shutter speed to prevent camera shake based upon the lens in use, but not always.

Since the camera doesn't know what type of photo you're actually making (e.g. portrait, landscape, or sports) the settings it chooses in auto or program modes aren't necessarily the best possible settings.  They are typically decent enough for the average snapshot, and a lot of people are fine with the results.  However, often there are better equivalent exposures to choose from that yield superior results that the more scrutinizing user will notice.

Note: An "equivalent exposure" is where two or more exposure result in the shadows, mid tones and highlights all being recorded at the same level.  For example, 1/500th @ f/8 has the equivalent amount of light (time and intensity) as 1/250th at f/11.  The shutter is opened for twice as long (1/500th vs 1/250th) but the aperture is cutting the intensity in half (f/11 is half the size of f/8).  Therefore if you were recording, say, a gray card, both 1/500th @ f/8 and 1/250th @ f/11 would produce exactly the same shot.

When you place the camera in one of these picture modes, you are telling it to choose an equivalent exposure that is more tailored for that specific type of shot.  Lets say you put the camera in full auto and the camera decides that 1/125th @ f/8 is the "correct" exposure.  Next, put the camera in sports mode.  In this mode the camera now knows that you're taking a photo that requires fast shutter speeds.  Instead of 1/125th @ f/8, it might now will say something like 1/500th @ f/4 which an equivalent exposure to 1/125th @ f/8.  F/4 is allowing 4x more light but 1/500th is 1/4 the time.

When you place the camera in landscape mode, the camera now knows that you are taking a landscape which requires a much larger depth of field than, say, a portrait or sports photo.  Therefore it will now use a smaller aperture.  Instead of f/8, the camera might use f/16 and a shutter speed of 1/30th to produce an equivalent exposure.  Instead of possibly not getting the foreground in focus (outside the depth of field) in full auto mode, you now are getting that area in focus.

Sports mode is similar.  The camera knows you're taking a fast action shot and will use the fastest possible shutter speed to stop the action.

Portrait mode will try to blur the background by using a very large aperture vs what it might have used in full auto.

A mode which is designed to use automatic settings to correctly photograph a scenic natural vista:

Lens set at wide angle. Aperture Priority mode at F11-F16.  ISO as low as practical. Single autofocus point 1/3rd of the way into the depth of the landscape (Hyperfocal Distance). Evaluative light metering. Image stabilisation enabled so that shutter speeds down to something like 1/30 second can still produce shake-free results.