You would mostly commonly hear the term eye of a storm when someone is speaking of a hurricane or tropical storm.

A hurricane is a low pressure system, which rotates counter-clockwise in the Northern Hemisphere. The "eye" is the center of the rotation, where the winds are calm.

Non-tropical low pressure systems do not develop an "eye" like a tropical system does. This is because the air in the center of the hurricane sinks, but in the case of a non-tropical low, the air in the center rises. Sinking air causes clouds to erode, whereas rising air causes clouds to form.

If you are unfortunate enough to ever be in the eye of a hurricane, you would experience very strong winds as the eye approaches, followed by a period of clear skies and calm winds while the eye is overhead. As the eye moves away, the wind would begin to blow very hard from the opposite direction.

Most tornadoes also rotate counter-clockwise and are considered mini low-pressure systems. Therefore some people may refer to the center of a tornado as the "eye of the storm", but I doubt many people have ever been there and lived to tell about it.

A well-defined eye is easy to see in this satellite image of Hurricane Floyd (September 1999).