An earthquake is a sudden movement of the Earth, caused by the abrupt release of strain that has accumulated over a long time. For hundreds of millions of years, the forces of plate tectonics have shaped the Earth as the huge plates that form the Earth's surface slowly move over, under, and past each other. Sometimes the movement is gradual. At other times, the plates are locked together, unable to release the accumulating energy. When the accumulated energy grows strong enough, the plates break free. If the earthquake occurs in a populated area, it may cause many deaths and injuries and extensive property damage.
Florida has had earthquakes, but they are rather infrequent and are not known to be severe. A 6.0 magnitude earthquake occurred about 260 miles off of Florida's west coast in September, 2006. Fortunately, there were no injuries or fatalities. Compared to California and the west coast of the U.S., the seismic hazards in the East are not well-understood. So, it is obviously possible for Florida to have a damaging earthquake, but it's pretty unlikely. In the last 200 years, very large earthquakes have occurred in Missouri and South Carolina. The Missouri earthquake was large enough to reroute the Mississippi River and the South Carolina earthquake was felt as far away as New York City and was strongly felt in Florida.
The USGS states that several minor shocks have occurred in Florida, with only one causing damage, St. Augustine January 1879:
Near St. Augustine, St. Johns County, Florida
1879 01 13 04:45 UTC (local Jan 12)
Plaster was shaken down and articles were thrown from shelves at St. Augustine and, to the south, at Daytona Beach. At Tampa, a trembling motion was preceded by a rumbling sound. Felt from a line joining Tallahassee, Florida, to Savannah, Georgia, on the north to a line joining
Punta Rasa and Daytona Beach, Florida, on the south. Two shocks occurred, each lasting 30 seconds.
Major earthquakes in the vicinity of Florida have also sent shocks into the state including the Charleston, South Carolina earthquake of 1886.
More information regarding Florida earthquakes can be found at the US Geological Survey