From Wikipedia on how to create a tesseract:
0 – A point is a hypercube of dimension zero.
1 – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one.
2 – If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square.
3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube.
4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract).
So far, this makes sense. However, the resulting image looks somewhat like this: (Source)
Given the "Instructions" on how to create a tesseract/hypercube, we see that with each dimension added, the dimensions of the cube's properties increase by one. We can further notice that the dimension of the cube's volume is always one higher than the dimension of the face (like we have wih regular 3d-cubes with a 3-dimensional Volume, 2-Dimensional Face and 1-Dimensional Edge).
So if we were to add another dimension, we'd have 3-dimensional Faces. To this point, the image is correct. However, the volume which should be 4-dimensional is shown as 3D, too. I know we cannot properly draw 4D-space, but none of the tesseract illustrations I have ever seen even mentioned this.