Mark Frauenfelder mentioned that the first time he saw me I was carrying a cardboard model of an "unfolded hypercube"---so I rooted around the house and, with my wife's help, found such a model, this one dates back to 1983.
I think it was the hyperdimensional mathematician Tom Banchoff who told me how to make this model. You cut out 28 cardboard squares and tape them, four at a time, to make seven partial cubes. These partial cubes have no tops or bottoms, they're like square tubes. And then you tape the seven partial cubes together as shown in the photos, making a very cool shape.
[You may notice that the shape is a bit like a hyperdimensional crucifix. Indeed, if I'm not mistaken, the artist Salvador Dali actually consulted with Banchoff when Dali did his well-known painting, Crucifixion (Corpus Hypercubus).]
It's a universal joint, in that you can swivel the top part freely---which is a little surprising as its all made of straight hinges. To make this into a real hypercube, you'd fold the band of projecting cubes to match the top cube and then---this is the hard part---you "fold" the bottom cube so its faces stretch around the outsides of the other cubes. The Wikipedia Tesseract page has a little animation that helps you visualize how this might work.
The idea behind the coloring on my model is that you start with one corner of the hypercube gray, and you think of the four dimensions as adding the colors red, blue, yellow, and white, coloring the successive 15 corners accordingly. I got the coloring pattern on this model from the early mathematician and science-fictioneer Charles Howard Hinton. In 1980 I edited a book of Hinton's amazing writings, Speculations on the Fourth Dimension. You can find a lot of this book (minus my introduction) online for free.
The letters on my model have to do with the fact that this particular unfolded hypercube was a gift to my wife on our sixteenth wedding anniversery. Our family members are S, R, G, R', and I---so you can start with S and think of each of the four dimensions as adding an R, G, R', and/or I.
One way to study a cube is to slice it into 2D cross-sections, taking the slices at various angles. By the same token, you can study hypershapes by slicing them into 3D cross-sections. Mark Newbold has written a nice Java applet called "HyperSpace Polytope Slicer" that lets you look at 3D cross-sections of four-dimensional shapes (or polytopes) like the hypercube. To use this applet, go to the link and click on the Controls button to get some interactive controls you can play with. Click on View to switch from a double view to a single view. (And better not click on Detach---at least on my machine, that often freezes up the applet.)
If you crave still more hypercube fun, I have two Windows progams written by my Master's Degree students, available for free download...including a 4D "Hyperspace Invaders" game.