For our needs there are three types of cardboard that matter the most: corrugated fiberboard, paperboard, and mat board.

Corrugated Fiberboard, a.k.a. 'Cardboard Box'

This is the cardboard you'll get with most shipping boxes, including your Adafruit packages! Typical corrugated fiberboard is made of three ply construction -- two smooth outer faces with a wavy, fluted corrugation layer sandwiched in the middle.

By Richard Wheeler (Zephyris) - Taken from http://en.wikipedia.org/wiki/File:Corrugated_Cardboard.JPG, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=6534720

Corrugated fiberboard is excellent for project construction because it is easy to cut, bend, fold, and drill, yet has quite a high strength to weight ratio. This strength is due to the stable arches inherent in the fluted corrugated inner wall.

Here are some primary types you'll encounter:

Single face board
Single wall double face board, a.k.a. '3 ply'
Double wall board with AB flute, a.k.a. '5 ply'

Just like a piece of wood, corrugated fiberboard has a "grain" to it. This is due to the orientation of the corrugation layer. The fiberboard is much more resistant to bending against the grain and easier to bend with the grain.

Thicknesses

There are a number of different thicknesses of corrugated fiberboard made. The most common are designated by the fluting letters 'A', 'B', 'C', 'D', 'E', and 'F'. The most common, used in most typical shipping boxes, is 'A' flute, which is 3/16" (4.8mm) thick.

The thin cardboard in the Nintendo Labo kits, for example, is E-flute which is 1/16" (1.6mm) thick.

By Chris 73 / Wikimedia Commons, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=2606709

Paperboard, a.k.a. 'Carton'

Also called 'chipboard', this is the single ply cardboard you find at the back of a notepad in a thick form, or making up your cereal box in a thinner form.

Paperboard doesn't have the strength of corrugated fiberboard, but it is much easier to bend and fold, which makes it ideal for forming small "papercraft" boxes.

Paperboard is also used to make tubes, such as those found in wrapping paper, paper towel, and toilet paper rolls.

Mat Board

Mat board is the material used in picture frame matting. It is a multi-ply, coated fiberboard with at least one smooth side. Great for cardboard circuits, as explained in this blog post by Bunnie.