Composition is the placement or arrangement of visual elements or ingredients in a work of art, as distinct from the subject of a work.
In other words, it is HOW the subject is arranged within the frame of the photograpH -- NOT WHAT the subject is.
It is so important because it helps us to create more interesting and compelling photographs, and it helps us to convey our meaning more convincingly. Once you start thinking about composition in your photography, it is hard to stop.
Here is what another photographer has to say on the topic. And here are some examples of elements found in photographic compositions:
Pattern and Repetition
Light and Shadow
Leading Lines
Shallow Depth of Field
Worm's Eye View
Frame Within a Frame
Texture
Aerial View / Bird's Eye View
Rule of Thirds
