Although not the correct term, 'The Dominican' is the way many North Americans refer to the Dominican Republic. If you use the term 'The Dominican' whilst in the Dominican Republic, you can expect a quizical look, as 'a Dominican' is obviously a person, so 'the Dominican' is - who exactly?
'The Dominican Republic' is admittedly quite a mouthful when you want to quickly refer to the country, so a more common variation amongst english speakers is simply 'the DR'. Make sure you say the letters D and R, not as the word 'doctor'. Nice and quick to say, and everbody knows exactly what you are talking about!