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," Fishbowl LA, 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.