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
Civil and criminal penalties may result from a violation of the