Most storm systems do not have a clearly formed center or eye, with strong winds swirling around a relatively calm center, but hurricanes (strong tropical cyclones) do. When the eye of such a storm moves directly overhead, the storm's winds abate sharply and the sky looks clear. This image from Boston.com shows what a hurricane's eye looks like from above—and why it is such an apt name. According to Wikipedia, the eye of a strong tropical cyclone "is a roughly circular area, typically 30–65 km (20–40 miles) in diameter."
As Autoresponder points out, the eye of the hurricane is dangerous because inexperienced people may suppose that the storm has departed when in fact it will resume at full force as soon as the storm's eye passes by and its eye wall (which immediately surrounds the eye and contains the hurricane's strongest winds) moves overhead. I grew up in a city on the edge of the Gulf of Mexico and can well remember being instructed in elementary school not to go out in a hurricane that suddenly became calm and seemed to have dissipated—because that was a sign that the eye was passing overhead.
So technically, "eye of the storm" refers to a false calm at the center of a real or figurative storm (specifically, a strong tropical cyclone). However, many writers use it to mean simply the middle of the action, where the upheaval is strongest. For example (from "Citibank Getting Burned In Studio Confusion," in Adweek, November 13, 2008):
The legal fight is the latest byproduct of a worsening financial picture both globally and in Hollywood. The Dow fell for a third straight day, shedding 411 points. Citigroup is at the eye of the storm in the media and entertainment sector, which is one reason its shares dropped nearly 11% to $9.64, their lowest level in 13 years.
Clearly the point here is that Citigroup is in the midst of the actual tumult, not in a calm envelope at the center of a gigantic storm, untouched for the moment as everything around it endures the full fury of the storm's blasting winds and rain.
Update (9/10/16): Early instances of 'eye of the hurricane' and 'eye of the storm'
A late answer challenges the idea that "the eye of storm" originally referred to a place of relative calm in a terrible storm, arguing that this notion is a corruption (based on knowledge drawn from satellite photography of tropical cyclones) of the original idea of the eye as "the raging heart of the storm." I decided to check to see how far back the expressions "eye of the hurricane" and "eye of the storm" go, where they came from, and how they were originally understood.
The earliest Google Books match for "eye of the hurricane" turns up in St. Nicholas, volume 54 (1927):
One of the special wonders of hurricane is its "eye," known as the "storm center." This "eye" is a region of peace and quiet, sunshine and fair weather, till it moves from that particular spot, when—presto—a marvelous change occurs. Violent winds tear buildings to pieces, uproot mighty trees, and hurl freight-cars and automobiles about like jack- straws, the peace in the "eye" of the hurricane having really been the "lull before the storm" which we often hear about.
This is consistent with this very brief entry in Ebenezer Brewer, Dictionary of Phrase and Fable (1870):
Eye of the storm. An opening between the storm clouds. (See BULL'S EYE.)
Brewer appears to have drawn his information from Andrew Steinmetz, A Manual of Weathercasts: Comprising Storm Prognostics on Land and Sea, volume 2 (1866):
In the torrid zone there appears what is called a "Bull's Eye," a small cloud suddenly appearing, seemingly in violent motion, as it were growing out of itself and soon covering the entire vault of heaven, producing a tumult of the elements, which id the more frightful, the more unclouded was the brightness of the sky immediately before. This must not be confounded with the "Eye of the Storm," which is merely an opening between the clouds.
Whether Steinmetz is talking about the central eye of a hurricane that is typical of such storms or merely about a sunlit rift in the storm clouds occurring by happenstance, it seems clear that the eye itself is not the most ferocious part of the storm.
Even earlier, Charles Tomlinson, The Tempest: an Account of the Origin and Phenomena of Wind, in Various Parts of the World (1861) explains the origin and sense of the term "eye of the storm":
The cyclone-disc is sometimes so thin at or near the centre, whether calm or not, that it may often be seen through, which clear space at the centre is termed by the Spaniards the eye of the storm. Thus in a cyclone in the Arabian Sea, in which the ship Seaton was dismasted, the account states that "during the height of the storm the rain fell in torrents; the lightning darted in awful vividness from the intensely dark masses of cloud that pressed down, as it were, on the troubled sea. In the zenith there was visibly an obscure circle of perfect light of 10° or 12°." In the whirlwind of Pacquebot des Ners du Sud, it is stated as a very remarkable fact that "while all round the horizon was a thick dark bank of clouds, the sky above was so perfectly clear that the stars were seen, and one star shone with such peculiar brilliancy above the head of the fore-mast, that it was remarked by everyone on board." In April, 1840, the Tigris encountered a short but severe cyclone in lat. 37° to 38° south, long. 68° to 75° east, and in the midst of it, while lying to, "the clouds broke away and the sun shown out, the whole surface of the water as white as snow with foam, and coloured like the rainbow in all directions. At 11 the wind blew with such fury that the three topgallant-masts were blown away, the spencer split to pieces, and furled sails blown to shreds from the yards."
It thus appears that the English "eye of the storm" was a borrowing from Spanish by 1861, and that the "eye" referred to the visible sky at the center of the storm—which is also the place of greatest (temporary) calm in a hurricane, as my earlier answer explains and as mariners in bygone centuries who survived passing through the center of such a storm could hardly have failed to notice.