The family of carbon nanotubes is large. There can be single walled carbon nanotubes (SWNTs), which are like a rolled up sheet of graphene - a monolayer of carbon bonded into a tubelike structure or they could be multiwalled carbon nanotubes, which have concentric layers of these graphene tubes. These carbon nanotubes can have large aspect ratios (length to width) or could be cut into ultra short carbon nanotubes. In fact, carbon nanotubes come in thousands of different molecular weights and isomers.
One of the most interesting characteristic of Carbon Nanotubes is how dependent the material properties are on small changes in structure. For example, small changes in the way that the carbon atoms align results in the difference between the SWNT being a metal or a semiconductor. This difference in the way the carbons align is called Chirality. Here is an easy way to demonstrate what chirality is. Take a transparency sheet with graphene's structure copied on it and connect two ends to form a cylinder. That is the model of one kind of carbon nanotube with the chiral index of (n, m) where n is the number of carbon atoms across the grid (at the center of each hexagonal structure on your transparency) and m would be zero since you havent moved down the matrix. If you want to make another kind of nanotube, you need to twist the graphene transparency and make a new tube that has a constant diameter. Scientists discovered an odd trend. When (n-m) is divisible by 3 (the product is an integer) then the SWNT is metallic, otherwise it is a semiconductor. Small changes in the arrangement of carbon atoms affects the electronic nature of these materials.
Make your own Carbon Nanotube by printing out this graphene sheet and trying different rollups. Notice how the diameter changes.