Let's start at the beginning:
A point has zero dimensions.
Move that point side-ways and it draws a one-dimensional line.
Next, drag the line (orthogonal, or 90-degrees, to the first motion) and it will create a two-dimensional square.
Then, pull the square up (orthogonal from both previous motions) and it will create a three-dimensional cube.
But we're not done.
Pull the 3-D cube into a fourth dimension (orthogonal to the other three) and it will create what is called a 'hypercube'. Each of the 8 corners moves along a new axis to create a new corner in the 4th dimension. So you can probably understand that a hypercube has 16 corners (or vertices).
Also, if you count the edges (12 in the original cube, 12 in the new cube, and 8 new edges linking the old vertices with the new vertices) it equals 32.
Have a look at the chart on Wiki and you'll see that our math is correct.
Link= Wiki on Hypercube
Also, enjoy the many Hypercube movies on YouTube.
Link= Hypercube on YouTube
We won't go past 4-D because the hypercube (or tesseract) is one of the simplest non-trivial polytopes. And E8 is the most complicated.
Once you see and understand the 'motion' of the hypercube, it is a wee bit easier to at least trust what is going on with the E8 movies.