A 0-cell is just a dot.
A 1-cell is a line segment; it's obtained by moving a dot (0-cell) through a small distance, which we'll take to be 1 unit. Here's a picture:
A 2-cell is a square; it's obtained by moving a line (1-cell) through a small distance perpendicular to itself. Here's a picture:
A 3-cell is a cube; it's obtained by moving a square (2-cell) a small distance perpendicular to itself. Since the screen you're looking at is just 2-dimensional, we have to represent a cube by "flattening" it out, or projecting it into two dimensions. Here's a picture of this kind of representation of a 3-dimensional cube in 2-dimensional space:
A 4-cell is called a tesseract. It is obtained by moving a cube (3-cell) a small distance in a direction perpendicular to itself. Since we can't draw anything that is perpendicular to the three dimensions we have already used for a cube, we draw below a kind of 3-dimensional representation, flattened into 2-space, of a tesseract. The same way we can look at a regular cube from many different directions (or rotate it in many different ways), there are many ways of looking at a tesseract. Here is one of them (the red arrow indicate the movement of the original cube "perpendicular to its three dimensions"):
You
can see the original cube here, since it is left with its edges as thick and
solid lines. The red arrows on each of its vertices indicate how that vertex
is being moved into a "4th dimension" perpendicular to the three occupied by
the cube. Each vertex of the cube traces out a new edge (of the tesseract).
Each edge of the original cube "sweeps out" a new 2D face of the tesseract.
Each face of the original cube now "sweeps out" a 3D "face" of the tesseract.
You can think of the tesseract as the 4D object "swept out" by the 3D cube as
it moves in a 4th dimension.
Here is the tesseract with all edges and vertices drawn the same way:
