Start with the line (1D). Make a square out of 4 of those, and you've got a 2D shape.
Take it into 3D but making a regular shape where each face is a square. You've got a cube.
A 4D form would be such that every face was a cube, and a 5D where every face was one of the 4D shapes, and so on.
A 4D version of the cube is called the Tesseract, and a fine example can be found at the link below.