Start with a square! As always, you can use any size square you like, but if you're new to origami, I'd suggest using a larger piece of paper for this project.

Crease some guidelines...

...so you can fold the corners into the center.

Then fold those corners into the center.

And once more, corners to the center again.

Then unfold it to the point where you just have one "corners to the center"

And flip it over.

Then, following the creases you made before, fold the corners in to the center in the opposite direction you did the first time. Then unfold them again.

Take the center point of the edge of one of the sides, and fold that in to the center.

Do the same with an adjacent side.

Squish the protruding square flat, following the creases that should be in the paper.

Repeat for the other sides.

Turn it over now. You should see four triangles on the back...

Fold these out flat. They will be your tabs. You'll use them to connect this module to the others in the cube.

Now turn the piece over to the front again.

Fold the inner corner of one of the squares out to the outer corner.

Repeat for the others.

Now the assembly...

This is actually the easiest part of the cube. Just put some tape or glue onto the tab that will go under...

Put the tab from the other piece right over the tab from the "under" piece, and stick.

Ta-daaa!

Possible uses:
Make this in pretty colors and hang it in front of a window
Save it for the holidays to hang on a Christmas tree
Bounce it!
Have your kitty chase it as you bounce it -- just make sure she doesn't eat it.
Be a nerd and change it up so you get said companion cube! (send me pics if you do!)