Update as of September 2023
Everything changed in the last 8 years. Nothing said below applies anymore.
Update as of March 18, 2015
Restrictions for US citizens traveling to Cuba have been relaxed even more:
(...)While tourism is still barred by statute, the new rules amount to permission for any American who wants to travel to Cuba to plan an educational sojourn there, as long as they keep records of their activities for five years.
Travelers who fill their days with museum visits, cultural sightseeing and conversations with Cubans about their society, and keep a daily journal, could meet the requirements. American officials suggested that there would be little policing of the comings and goings of those making people-to-people trips.
Update as of January 15, 2015
In December, 17 2014 Obama announced modifications to the travel regulations and the embargo against Cuba in general. The details can be found in the following page:
Fact Sheet: Treasury And Commerce Announcement Of Regulatory Amendments To The Cuba Sanctions
Summarizing, although it is still technically illegal for US citizens to visit Cuba as regular tourists, the restrictions have been relaxed and there are a lot more of possible scenarios that make it possible to travel there, for example it is not longer required to apply for an explicit permission, traveling with a specialized company that has a permit is enough. On top of that, US credit cards can now be used in Cuba, and US citizens are now allowed to spend money in Cuba.
There is an ongoing debate in the US government currently about whether to lift the embargo entirely or not.
My old answer:
As far as I know, visiting Cuba without a permission from U.S. Department of the Treasury is illegal.
Title 31--Money and Finance: Treasury
CHAPTER V--OFFICE OF FOREIGN ASSETS CONTROL, DEPARTMENT OF THE TREASURY
PART 515--CUBAN ASSETS CONTROL REGULATIONS
(I wonder if posting this information here is also illegal...)
Even if you do have permission, some restrictions still apply, as stated in this document (CUBA TRAVEL ADVISORY):
Authorized travelers to Cuba are subject to daily spending limits and are prohibited from bringing any Cuban “souvenirs” or other goods into the United States, with the exception of information and informational materials.
Civil and criminal penalties may result from a violation of the Regulations.
