On July 10, 1908, a complicated apparatus working in the laboratory of Heike Kamerlingh Onnes in Leiden, Holland, managed to produce 60 ml of liquid helium, at a temperature of 4.2 Kelvin, or −269°C.
Johannes Diderik van der Waals on the equation of state of real gases. He had been scooped in the liquefaction of hydrogen (at 20.3 K) in 1898 by James Dewar (who, in the process, had invented the Dewar flask).
But as it turned out, the liquefaction of helium required a multi-step strategy and a big laboratory, and this was Kamerlingh Onnes' business: Using first with liquid air, then liquid hydrogen, helium could finally be cooled enough, via the Joule-Thomson effect, to condense into the liquid state. The physics laboratory in Leiden had become the "coldest place on Earth", and immediately turned to the international centre for low-temperature physics.
Three years later, in 1911, Onnes found that mercury lost its electrical resistivity when cooled to the temperature of liquid helium - this was the discovery of superconductivity. In 1913, Kamerlingh Onnes was awarded the Nobel Prize in Physics, "for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium".
I read about the story of the liquefaction of helium in the July issue of the PhysikJournal (the German "version" of Physics Today - PDF file available with free registration). Moreover, the Museum Boerhaave in Leiden shows a special exhibition to commemorate the event, "Jacht op het absolute nulpunt", but the website seems to be in Dutch only. However, the curator of the exhibition, Dirk van Delft, describes the story in a nice article in the March 2008 issue of Physics Today, "Little Cup of Helium, Big Science", where he makes the point that the Kamerlingh Onnes Laboratory in Leiden marked the beginning of "Big Science" in physics (PDF file available here and here).
One Hundred years later, there is a twist to the story I wasn't aware about at all: Helium is now so much used in science and industry that there may be a serious shortage ahead! 
Helium Demand ...
The following graph, plotting data provided by the US Geological Survey, shows how helium is used today in the US:
The biggest chunk of helium is used for technical applications, which include pressurizing and purging, welding cover gas, controlled atmospheres, or leak detection. The second-largest part is already usage in cryogenics, such as in the cooling of superconducting magnets for magnetic resonance imaging (MRI, formerly known as nuclear magnetic resonance, NMR) machines in medicine, and of superconducting cavities and magnets for high-energy particle accelerators. Only then follow applications that include lifting, as in balloons or blimps.
The LHC, for example, needs 120 metric tons of liquid helium to cool down the accelerator to a mere 2.17 Kelvin, when helium becomes a superfluid and an ideal thermal conductor (90 tons are being used in the magnets and the rest in the pipes and refrigerator - see p. 33 of LHC the guide), and 40 more tons to cool down the magnets of the large detectors to 4.5 Kelvin, so that the coils are superconducting . But even this huge amount of helium is just about 5% of the annual US consumption of helium for cryogenics!
...and Helium Supply
Helium is the second-most abundant element in the Universe, but on Earth, it is rare: The atmosphere cannot hold back the light noble gas atoms - ionized helium is transported along magnetic field lines into the upper atmosphere, where it's thermal velocity exceeds the escape velocity of 11.2 km/s .
Thus, the constant helium content of about 5 parts per million (ppm) in the atmosphere is maintained only because helium is constantly being produced anew in radioactive decay: for each uranium, thorium or radon nucleus undergoing alpha decay in the Earth's crust, a new helium atom has emerged. This helium gas accumulates in gas fields within Earth, often together with natural gas. That's where helium can be won.
The following figure compares the annual helium production in the US from the exploitation of gas fields with consumption and exports, and with the total World production (data according to the US Geological Survey, who is to blame for the anomaly that the US production can exceed world production):
US helium consumption and exports clearly exceed production, which is possible because the US helium stock is being consumed. World helium production is still raising at the moment, but easily exploitable reservoirs will become rare some time in the future, as they are already now in the US.
Fortunately for future particle accelerators, and all other applications of helium in science and technology, helium can also be won back from the atmosphere, albeit at a higher cost:
Walther Meissner succeeded to produce liquid helium in Berlin in 1925 , he could not rely on helium supplied from american gas fields because of embargoes in the wake of World War 1 - helium to fill balloons and Zeppelins was considered of highly strategic value. Instead, he cooperated with a company who later sold commercially equipment to liquefy helium. And he could distill enough liquid helium to discover, together with his postdoc Robert Ochsenfeld, that superconductors expel magnetic fields – the Meissner-Ochsenfeld effect.
 For the pending helium shortage, see for example
- The coming helium shortage, by Laura Deakin: "It’s surprising how many scientists and nonscientists alike are oblivious of the pending helium shortage. But it is a fact—we will run out of helium. [...] The question is when, not if, this will happen." (Chemical Innovation 31 No. 6, June 2001, 43–44)
- Helium shortage hampers research and industry, by Karen H. Kaplan: "If new sources of helium aren't developed, the world's supply of the gas will dwindle and prices will soar." (Physics Today, June 2007, page 31)
- Helium Supplies Endangered, Threatening Science And Technology: "In America, helium is running out of gas." (ScienceDaily, January 5, 2008)
 For the cooling of the LHC, see for example
- Let the cooling begin at the LHC, by Hamish Johnston: "Tens of thousands of tonnes of equipment must be cooled to near absolute zero before the Large Hadron Collider can detect its first exotic particle. The head of CERN's cryogenics group, Laurent Tavian, tells Hamish Johnston how this will be done." (Physics World, November 7, 2007)
- Messer to provide helium for LHC project, by Rob Cockerill: "Over the course of the next few years, industrial gas specialist [...] is to provide a 160.000 kg supply of helium to the European Organisation for Nuclear Research (CERN) for the operation of the world’s largest particle accelerator." (gasworld.com, January 23, 2008)
- Cern lab goes 'colder than space', by Paul Rincon: "A vast physics experiment built in a tunnel below the French-Swiss border is fast becoming one of the coolest places in the Universe." (BBC News, July 18, 2008)
- Cooldown status - the current state of the cooldown of the LHC, from CERN.
 See for example page 250 and 251 of Noble Gas Chemistry by Minoru Ozima, Frank A. Podosek, Cambridge University Press, 2002.
 Verflüssigung des Heliums in der Physikalisch-Technischen Reichsanstalt, by Walther Meissner, Naturwissenschaften 13 No 32 (1925) 695-696.