Content-Aware is a technology that enhances the capabilities of multiple tools within Photoshop that assists in patching, (re)moving, scaling, filling, and extending pixels in rasterized images. It shows up in various tools throughout Photoshop but the technology behaves according to which tool it's being applied with.
The most basic explanation: when activated, it recognizes the content in your image to determine how to best make up the differences made in editing processes.
For example, when applied to the Patch Tool:
The Patch tool is used to remove unwanted image elements. The
Content-Aware option in the Patch tool synthesizes nearby content for
seamless blending with the surrounding content.
-Adobe Support:Content-Aware Patch and Move
And in regards to the Move Tool:
Use the Content-Aware Move tool to select and move a part of a
picture. The image is recomposed, and the hole left behind is filled
using matching elements from the picture. You do not need to carry out
intensive edits involving layers and complex selections.
You can use the Content-Aware Move tool in two modes:
Use the Move mode to place objects in different locations (most
effectively when the background remains similar).
Use the Extend mode
to expand or contract objects such as hair, trees, or buildings. To
best extend architectural objects, use photos shot on a parallel
plane, rather than at an angle.
-Adobe Support:Content-Aware Patch and Move
These are just a couple of examples of how the Content-Aware technology is applied to different tools within Photoshop. As you can see, it's a powerful editing enhancement that uses the pixel information from the rasterized image to best guess how to fill in the "holes" that result from various editing operations. For a more in depth look at all of the Content-Aware applications, Adobe Support has them here.