We live in a 3D world, so 3D Cartesian coordinates are used to describe problems in our world. However, until we have holographic displays, the way we normally display information is 2D, such as paper or monitors. So 3D is normally projected onto 2D; this could be a perspective projection or orthographic projection.
The most common uses of 3D coordinates are in engineering, physics, and computer graphics. This would be done with software such as AutoCAD, Maya, SoftImage, 3DS Max, and Blender. As you know, computer animated movies can be shown in 3D or 2D, but both still have perspective. In order to have perspective projections, it is required to model everything in 3D.
Motion capture is done in 3D, so that they can be put in computer games or special effects in movies as well as the animated movies.
In engineering, buildings or bridges or any structure, require 3D so that workers know how wide, long, and high to build things. Another use that you may have heard about is 3D printers; obviously you need 3D coordinates to send to the printer for it to build anything meaningful.
There are some other uses that you may know such as topographical maps that show the height of a 2D position or an air traffic display will show the 2D position of an aircraft, but display its altitude beside the position.
These are just a few of many examples.