What is meant by the 3-D Cartesian coordinate system?
For much of math, we speak about a 2-D Catersian coordinate system with just the (x,y) components. In that system, you can only go horizontally in the x-direction a certain amount and vertically a certain amount in the y-direction. With the 3-D Caterisian coordinate system, it's taken one step further- There is a new direction, the "z" direction, which indicates a more realistic coordinate system. The three dimensions of x,y, and z indicate positions of lines/points that can actually be modeled in the real world.
To better visualize this, you can think about a page representing the 2-D coordinate system (ignore the thickness of the page so that it seems 2-D). On that page, you can only draw lines that lie directly on that page. They have only the dimensions of x and y. However, with 3 dimensions, the "z"component can indicate whether that point is situated "into" or "out of" the page. Imagine poking a thin stick through the page. That could indicate your z axis, which adds a whole new dimension to the illustration.
This is better explained with a picture. This is a picture of a typical Cartersian Coordinate system:
Note that the orientation of the x-y-and z axes can change according to convenience of use, but these are just to illustrate how the 3-D factor of the Cartesian system comes into play. There are three axes to define position with a 3-D coordinate system, unlike the singular xy plane in the 2-D coordinate system.