Why graphene is interesting from a physicist's point of view:
(i) In graphene the electrons contributing to an electrical current behave as they do not have a mass. This is special to graphene and some other novel materials. (First discovered in graphene)
(ii) The conductance of graphene, although a semi-metal, can be tuned using the field effect. This is how usual transistors work. Why are there no metallic transistors? Because we can not tune the conductivity by electrical means (so simple). This remarkable property of graphene arise from its two-dimensional nature. It is so thin that electric field is not screened by conduction electrons. Because of this, graphene is expected to play a dominant role in thin-film transistors, the transistors that are so thin that they are transparent and could be used on screens. Also, graphene-based devices can be flexible and so will probably be utilized in wide range of applications.
(iii) In graphene, the electrons contributing to a current are not scattered during flowing from one point to another as much as in other materials. That is, the mobility of the electrons are very high. (conductivity = mobility x electron concentration) This is because the scattering (what causes resistance as we know it) is suppressed owing to the chirality of the conduction electrons of graphene: they have to change their pseudo-spin to scatter away from their trajectories. And electrons do not like that.
What brought graphene a Nobel Prize in phyiscs is that after the first study on graphene, all the laboratories started to investigate this material. This is mostly because:
(i) It is so easy to fabricate graphene-based devices as pointed out in this thread by other people as well. You can obtain a graphene using the so-called mechanical exfoliation method. Why they came up with such a cool-sounding complicated name is no one was fond of saying "oh, we got it using a scotch tape."
(ii) From an engineering (practical) point of view, graphene is really promising in electronic applications. So the funding was quite well.
The physicists K. Novoselov and A. Geim were awarded for their first paper about graphene: "Electric Field Effect in Atomically Thin Carbon Films" (http://www.sciencemag.org/content/306/5696/666.short) Check how many citations this article has. It has been less than a decade.
Related carbon-based nano-materials:
Buckyballs are molecular spheres made of 60 carbon atoms. Due to their nature, they are not particularly useful for transport measurements (where you apply a voltage and measure the current flowing), because it is not easy to fabricate contacts on a tiny sphere. (I have experience only in transport measurements so I don't know much about them.) But a little more than a decade ago, they were utilized in matter-wave interference measurements. Remember the particle-wave duality? For example, electrons are particles but they can also interfere as if they also have a wave-like nature. The experiments on electrons demonstrating the electron-wave interference have been performed in early 20th century. From the pioneering work of French physicist De Broglie, we now know that there is a particle-wave duality. And buckyballs showed also a wave-like nature! They made two beams out of buckyballs, they made the beams collide and obtained an interference pattern. This experiment was performed in 1999: "Wave–particle duality of C60 molecules" (http://www.nature.com/nature/journal/v401/n6754/abs/401680a0.html) Currently, people are trying to do similar experiments on viruses; even larger particles. Maybe in the future we will obtain an interference pattern also out of living things such as bacteria or so. Or maybe cats! Who knows?
Carbon nanotubes (CNTs) are just rolled up graphene. Also from a quantum mechanical point of view, they are treated as such: To obtain the electronic properties of carbon nanotubes, we start from a graphene sheet. Then we roll it up. This is equivalent to the Bloch problem, the problem how we now calculate the energy bands of crystals. Of course, you can roll a graphene in many ways. This can result in CNTs with different diameters and with different orientation of carbon bonds along the tube axis (called helicity). The diameter and the helicity of CNTs determine their properties. This way CNTs can be metallic or semiconducting. (Note that graphene is a semi-metal.) CNTs are very promising for a large range of applications. First, they are really really thin, even though they are mechanically extremely robust. This we all have probably already heard of. They also share some similarities with graphene. So for example scattering in CNTs are also suppressed because of the pseudo-spin of the conduction electrons. This pseudo-spin is locked to the movement direction of the electrons, so they can not change their direction of propagation without changing their pseudo-spin. That is, the scattering is suppressed. However, the future of CNTs in electronic applications might be in danger. This is because even though there has been quite some time since the discovery of CNTs, we still can not control the diameter and the helicity of CNTs: We can not produce arbitrary amounts of identical CNTs. And this is of course a big big challenge for industry. Because of this, their popularity in transport measurements are decreasing. Not so many labs are investigating CNTs as they did 10 years ago.
