Cohiba is indeed a very famous brand of Cuban cigar, but many of the Cohibas on the market today are not Cuban. Here's the scoop.

First of all, the brand is surprisingly new: the first Cohibas under that name were only produced in 1968, 9 years after the Cuban Revolution, for the exclusive use of Fidel Castro and his guests at that. However, the tobacco plantations and cigar factories that produce what is today Cohiba have been around for a lot longer, so others can with some degree of legitimacy claim to retain the heritage of the "real" Cohiba.

Cuba

Cuban Cohibas are still made in Havana, Cuba. This is the "original" Cohiba, in the sense that this is the location where the brand was developed, but during the Castro era the quality of the cigars has, in the opinion of many, gone down from the golden days. This has been noted as a problem in Cuba itself though, and quality control has improved quite a bit lately.

  • Identification: Gold, black and white label with the words "Cohiba" in block print and "La Havana, Cuba" in script underneath. The black label has precisely 4 rows of white squares (not circles), three of which extend above the "Cohiba" text.
  • Warning: This is the most counterfeited cigar in the world. Buyer beware. And, of course, importing them into the United States is illegal.
Dominican Export

Export Dominican Cohibas are made in the Dominican Republic by cigar experts who left Cuba after the revolution. However, because the island and its tobacco is different, the taste is also different from Cuban Cohibas. Whether different means worse is a value judgement, and these are generally regarded as excellent cigars as well. The brand is owned by General Cigar Holdings Inc and is for export only; you cannot buy these in the Dominican!

  • Identification: Black, cream and red label with the word "COHIBA©" in raised lettering in the center and "HAND MADE" on the right; often called "red dot" Cohibas because the O is filled in red.
Dominican Domestic

But there's more: the copyright to the name "Cohiba" in the Dominican Republic for sale only in the Dominican is held by an unrelated operation! These may not be sold in (or to) the United States, but you can freely import them for personal use from the Republic. However, these are generally regarded as the worst of the bunch, so you'll probably want to steer clear.

  • Identification: Otherwise identical to the Cuban Cohiba label, except that "La Havana, Cuba" is replaced with "Republica Dominicana".
Summary

Wasn't that fun? So if you're going to go shopping for Cohibas, I'd suggest careful examination of Cigar Aficionado's counterfeit gallery:

http://www.cigaraficionado.com/Cigar/Aficionado/Counterfeit/countgal.html

And good luck: a real, fresh Cohiba is definitely one of the most enjoyable ways to get lung cancer.