A polyhedron is a 3dimentional solid that contains flat faces and straight edges. You may be familiar with the five shapes known as the Platonic solids: the tetrahedron (4sided pyramid), the cube, the octohedron, the dodecahedron, and the icosahedron. These are the polyhedra that can be constructed using one type of polygon for all of the faces, such as a triangle, square, or pentagon.
There are also the Archimedean solids, which are shapes that have faces of different regular polygons, such as a combination of squares and triangles. In addition, some of these polyhedra can be "stellated"  their sides can be extended until they intersect with other extended sides. The Great Dodecahedron is an example of a stellation of a dodecahedron.
I created the polyhedra on this page with standard U.S. business cards (2" x 3.5"), folded to create two trianglar faces per card. This is the unit shape from which any of these polyhedra can be built. The unit shapes can be combined to form triangular, square, pentagonal, and hexagonal polyhedral faces. Some of these faces are actually cumulations, where the face is replaced with a pyramid, with the (invisible) polygonal face as its base; the pyramid has either negative or positive height with respect to the face. For example, the dodecahedron seen below is actually a cumulated dodecahedron, because the faces are not truly pentagons, but pentagonal pyramids with negative height.
In this article, you'll learn how to construct any polyhedron that contains any combination of triangular, sqaure, pentagonal, or hexagonal faces. Here are some examples:
Icosidodecahedron  Dodecahedron  Small Rhombicuboctahedron 
 Standard business cards (2" x 3.5")
 Scotch tape (optional for some constructions)
 An idea of what you'd like to construct
All of the constructions on this page use the same basic shape. Here's how to fold it.
 Hold the card faceup. Fold the card so the top, right corner meets the bottom, left corner.
 Notice the parts of card that overlap. Fold these pieces back (actually, whether you fold the overlaps backward or forward
depends on the solid you'll be making), so the fold lines up with an edge of the card. Crease the fold. When you let go of the card,
you will have a shape that looks like this:
 That's it! Now make as many of these as you need for your solid.
The tetrahedron and octahedron are two of the easiest solids you can make. Both solids have triangular faces. Since each unit shape contains two triangles, you'll only need two units for the tetrahedron, and four for the octahedron.
These shapes will also stay together without the use of tape. It's quite satisfying to put the units together, because there's a stable state that occurs when the unit pieces are arranged properly and the shape comes together. It's as if all of the faces are exerting equal pressure on the other faces, supporting each other.
For both of these shapes, the overlap tabs will appear on the outside of the solid. Therefore, when you fold the overlaps, fold them in the same direction as the main diagonal fold (since the diagonal fold is a valley fold, the overlaps will also be valley folds). Then, put the pieces together. The overlap tabs of one unit will embrace the nontabbed edge of another unit.
The octahedron takes a little practice to hold together  with only two hands, it can be tough to bring four units together  but you can do it!
If you make six octahedra, we'll have fun with those later.
The icosahedron (20 faces) is the next logical step: it's the other Platonic solid with triagular faces, but it's a bit more advanced than the octahedron. With 20 sides, therefore 10 unit shapes, you'll have a hard time putting this together without the help of tape. You're welcome to try. Otherwise, use tape on the inside of the shape.
Start by putting five units together to form a pentagonal pyramid. Each triangle that comprises the pyramid will be one end of the unit shape. The other end of the unit shape will hang off, waiting to be connected to other pieces.
Next, create a second pentagonal pyramid. Finally, bring the two layers of extra sides together  the overlap tabs of one face will embrace the nontabbed edge of another face, and you'll have another Platonic solid for your collection.
There is an alternate way of creating the icosahedron: as a cumulation with pyramids of negative height. In this case, the units are folded tightly, and the tab of one unit nestles between the two foldedtogether faces of another unit. You won't need tape for this construction. Since there are 3 faces per pyramid and 20 sides, you'll have to plan for 60 faces, so you'll need 30 business cards for this solid.
Both ways of constructing the icosahedron will result in a solid with the same dimensions (except for minimal differences from the thickness of the cards).
Next stop: the dodecahedron (12 faces). Geometrically, the dodecahedron and the icosahedron are related in all sorts of interesting ways.
We'll actually create a cumulated dodecahedron, because the pentagonal faces will be represented by pentagonal pyramids. 12 faces with pyramids of 5 triangles... that's 60 trigular faces altogether, or 30 cards. (Hey, doesn't the cumulated icosahedron also have 60 triangular faces?)
You'll need tape for this shape, and you'll need to fold the tabs in the opposite direction of the main diagonal fold.
Here's a tip when assembling these pieces: If you look at the picture, you'll see that the diagonal fold of a unit is the edge between faces of the solid. The diagonal fold does not appear inside the pentagonal pyramid. Just something to keep in mind to make sure you're building the shape correctly.
To start, build a pentagonal pyramid with 5 units. Then, for each half of the unit that isn't part of that pyramid, build another pyramid around it  sometimes using unithalves from the neighboring pyramid (you'll know when).
I imagine you're getting the hang of this now, and you can probably figure out how to make pyramids with 4 and 6 triangles. The 4triangle pyramids will let you represent square faces, and a 6triangle pyramid turns out to be as flat as a plane.
Here are some of the other shapes I've made using this technique. After making the shape, the next bit of fun begins  going to Wolfram MathWorld to figure out, "Is there a name for what I just built?"
Stella Octangula
 
Icosidodecahedron
 
Octahemioctahedron
 
Truncated Octahedron
 
Small Rhombicuboctahedron
 
Spikey
 
Spikey

Since all of these solids are built with the same unit shapes, there are some interesting relationships that you can see between the shapes.
Imagine the stella octangula as two tetrahedra merged together. The volume that is common to both tetrahedra is an octahedron made with unit pieces.  
The end of an icosahedron (or a cumulated icosahedron with pyramids of negative height) fits within the pentagonal pyramid of the cumulated dodecahedron. (If you're really ambitious, build one dodecahedron and 12 icosahedra, then assemble them  you'll only need 90 business cards!)  
A 4sided pyramid can be filled with an octahedron. If you create 6 octahedra, you can plug them into an octahemioctahedron (as well as a truncated octahedron). The result? A larger octahedron! 
By now, you must be inspired to build your own polyhedra! If you create something cool, I'd love to showcase it here  and, of course, give you credit for your creation.
You can also create solids with different unit pieces. For example, you can create a Great Dodecahedron with unit shapes that use isosceles triangles with sides 1, 1, Φ instead of equilateral triangles with sides 1, 1, 1.