Molecular nanotechnology is the ability to build objects to atomically exact specifications; to synthesize complex structures by using precise, mechanically controlled positioning of individual chemical building blocks. Real nanotechnology, in the form originally envisioned, should not be confused with the modern misuse of the term to denote nanoscale bulk manufacturing - technologies operating on a scale of tens or hundreds of nanometers which do not involve atomic-scale precision or the exact specification of molecular structures. Real nanotechnology would be a world-shaking development, and the term shouldn't be appropriated to describe the latest computer chip; but unfortunately eager venture capitalists have now confused the issue beyond repair. The people still discussing real nanotechnology usually use the term "molecular nanotechnology" to avoid confusion; this term has not been abused as yet.
Although nanotechnology probably still lies years in the future, the theory has already been analyzed in considerable detail. One of the best references is "Nanosystems: Molecular Machinery, Manufacturing, and Computation". Every now and then a chemist or physicist whose knowledge of nanotechnology derives from overheard conversations at cocktail parties is quoted in the newspapers as declaring loudly that nanotechnology is a crackpot notion because, e.g., scaling laws mean that all the engineering rules are different at the molecular level. Well, yes, scaling laws do mean that the rules change at the molecular level, but the way in which they change has been analyzed in considerable detail. If anyone tries to tell you that nanotechnology is impossible, ask to see their math. Advance analysis of nanotechnology has long since moved beyond vague arguments to running the equations and, in a few cases, specific technical details of proposed systems.
So why all the controversy? Because, if and when molecular nanotechnology comes into real existence, the consequences will be enormous. We're used to the idea that computers get faster as they get smaller, but for nanotechnology "faster" doesn't even begin to cover it. For example, suppose that you build a nanocomputer using diamondoid rods of a few thousand atoms apiece as the computer elements; they click against each other, mechanically, and thereby implement transistors and logic gates - an extremely miniaturized abacus. Chapter 12 of Nanosystems brings together the pieces of such a system, analyzed in previous chapters, and goes on at length about energy requirements, heat dissipation, thermal excitation and error rates, acoustic signal propagation speeds, clocking based on oscillating drive rods... and so on, but at the end of the chapter we find the final numbers: A CPU-scale system containing a million transistors would fit within a 400-nanometer cube, run at 1GHz, and consume 60 nanowatts. A desktop nanocomputer consuming 100 watts of power would process 1018 instructions per second. That's a billion billion instructions per second, roughly a billion times faster than a modern Pentium chip, and an order of magnitude higher than most estimates of the power of a human brain. (1011 neurons and 1014 synapses firing 200 times per second ~ 1017 operations per second.) A faster nanocomputer might consume a hundred kilowatts, process 1021 operations per second - ten thousand brainpower - and still fit inside a single cubic centimeter. (Yes, you can cool it.)
These numbers should be taken with a large grain of salt, except that the grain of salt is in the opposite of the usual direction - a real nanocomputer would be faster, not slower. The design analyzed in Nanosystems has solid rods shuttling around at the speed of sound in diamond, not because rod logics are optimal for real systems, but because rod logics are easy to analyze. Preliminary numbers on electronic nanocomputers, published a few years later, suggest somewhere on the order of 1025 operations per second for a "desktop" system - roughly a hundred million brainpower - but nanotechnologists still stick to talking about rod logics because rod logics have been extensively analyzed and the assumptions are easy to defend. And yet rod logics are essentially the nanotechnological equivalent of vacuum tubes. Nanosystems contains a design for a six-degrees-of-freedom manipulator that would have used around 4 million atoms; a later, atomically precise design for a six-degrees-of-freedom manipulator turned out to require only 2,596 atoms. If the ultrafast nanocomputers that are usually analyzed are the equivalent of vacuum tubes, then the corresponding ultrafast manufacturing technologies are the equivalents of steam engines. The kind of nanotechnology we know how to analyze in advance, despite its enormous power by comparison with current technology, is a very awkward and primitive form of nanotechnology. This shouldn't be all that surprising; current human technology just isn't impressive in a cosmic sense - we're nowhere near the limit of what we already know is possible.
Nanotechnology often pops up in discussions of the Singularity for three major reasons. First, when the components are individual atoms, the world contains an unlimited supply of perfectly machined, perfectly interchangeable parts; molecular factories can reproduce exponentially. (Nanosystems concludes that a one-kilogram factory should be able to build an exact copy of itself from acetylene feedstock, plus some nutrients, in around a thousand seconds - i.e., less than twenty minutes. Closing the loop completely by extracting materials and energy from a natural environment is possible - biological organisms do just that - but would probably be rather unwise to attempt while we're still all human-level intelligences; there's this thing called the "grey goo" problem...) The benefits of the Singularity need not be absurdly expensive, even during the very first stages; like free software, self-replicating technologies can be created once, and then used by millions or billions for very little additional cost. Second, while a molecular manufacturing factory can build exact copies of itself, it can also build improved factories instead. The time it takes to go from the very first self-reproducing box to the most advanced technology that can possibly be built from individual atoms is twenty minutes plus however much time is required to do the thinking. A fast mind with nanotechnological "fingers" can build and test and observe on a scale where actions take microseconds, nanoseconds or even picoseconds. The time it takes to go from nanotechnology to the ultimate limits of material science - assuming there are any - could turn out to be effectively zero, at least from the standpoint of a modern-day human. Third, even the most primitive nanotechnology is still advanced by comparison to biology. It may take a mature nanotechnology to handle uploading, but for almost anything else, the steam engines and vacuum tubes that are usually analyzed should be more than enough. The characteristic size of a nanodevice is considerably smaller than the characteristic size of a cell (this mobile in vivo nanodevice design is 1 micron; an average cell is 10 microns), and the manipulators contained in a nanodevice are even smaller. To a nanodevice, a cell is a large, complex structure whose parts can be handled and manipulated. Almost all medical problems would become trivial tasks: extinguishing viral and bacterial infections, removing or even repairing cancerous cells, regeneration of missing body parts, the various challenges involved with aging reversal, and so on. Food could be synthesized directly from dirt and sunlight (plants do this slowly and inefficiently), which would be enough - in a material sense - to solve world hunger.
Actually, world hunger today is an essentially political challenge rather than a material or technological challenge; we grow enough food, we just can't get it through the various warlords holding the territory. This is why it's important to emphasize that the Singularity is not just a physical technology like nanotech; it involves smarter-than-human intelligence. Similarly, a "utility fog" might be able to intercept bullets fired at a living target - although this also might require mature nanotechnology - but from here to ending war is, again, more of a Singularity issue than a nanotechnological one. The applications listed above underestimate the impact of the Singularity, or at most, provide a lower bound; they describe a patchwork of mid-range technological solutions to current nagging problems, rather than an encompassing transition to a transhuman standard of living. An upload doesn't need nanotech medical treatment. Of course we have no idea what a post-Singularity way of life would really be like, but we can see that a patchwork of nanotechnological bandages isn't it.
From a Singularity perspective, the point of discussing nanotechnology is to say that if you can get control over the physical world at a molecular level, you can solve problems that exist at higher levels of organization. You can do a lot more than just solving those problems - reprogramming matter is overkill if all you're trying to do is cure cancer. But the nice thing about overkill is that it works.
Now consider the other points raised above: nanotechnology can self-replicate, and the characteristic speed of transistors and nanodevices is very rapid when compared to the thoughts and biology we know. The impact of the Singularity will not necessarily resemble the technological and economic revolutions we know - revolutions that took years or decades to complete. The Singularity may take the form of a literal wavefront sweeping out from a literal epicenter.
From the perspective of the Singularity, nanotech is the prototypical example of rapid infrastructure. Nanotech can self-replicate; it moves physical manipulation into a new timescale - microseconds instead of seconds - and thus acts as a very rapid base for the acquisition of further ultratechnologies; and it turns matter into information, allowing in a sense the "reprogramming of reality". Whether nanotechnology turns out to be important in itself, or only as a platform for reaching technologies as yet unimagined, nanotech can be said to be an informational challenge rather than a material challenge. We could almost certainly build nanotechnology with present-day tools; we just don't know how. We can synthesize arbitrary sequences of DNA, and we can turn arbitrary DNA strands into proteins; one floppy disk, if we only had the right floppy disk, could probably contain a DNA sequence that would create a protein that would act as a very crude manipulator that could be directed by acoustic information to create a diamondoid nanodevice which, though itself slow and crude, was flexible enough to create the most advanced nanotechnology for which the user possessed an atomic specification. It may take years for humanity to cover this territory, arriving at "wet" nanotechnology long before "dry" nanotechnology, and perhaps taking the pathway of larger-scale chemical synthesis or finer-scale mechanical manipulation. We lack the knowledge needed to take shortcuts. We still haven't cracked the protein folding problem; instead of using evolution's molecular factories, it may be easier for us to go down the long slow path of building our own molecular factories from scratch. But that would be a human story; a story of human-sized minds running at human speeds.
Nanotechnology may or may not be the final technology, and the future may or may not lie in countless diamond specks. Rather, nanotechnology represents the human paradigm of material technology pushed to its final limit - nearly complete power over the physical world. Since technology is the product of intelligence, we can expect nanotech or some such similar ultratechnology to be the product of the Singularity, but in a final sense the Singularity is unrelated to nanotechnology; the Singularity is the breakdown in our model of the future that occurs with the arrival of transhuman intelligence, not any particular increase in material capabilities. The point of discussing nanotechnology is to show that technologies accessible outside the human regime are significantly more powerful than those we are accustomed to dealing with; they can bypass existing human infrastructure; operate on timescales far quicker than human hands; and provide a substrate for thought tremendously faster than both existing human brains and existing computer technology.
So called "machine phase" nanotech is alive and doing very well. In fact, I have to confess that it is moving faster than I anticipated. In reference to the article specifically, the US does seem to be leading the field in machine phase nanotech. What you are talking about is "wet" nanotech, which while also moving along, isn't as easy to engineer as machine phase technologies.
That said, the article glosses over a lot of important points. This is not a simple topic, and can run far afield while still being germane.