All arriving visitors to the DR will need a passport for entry. Your passport must be valid for at least six months and it must have sufficient pages for entry and exit stamps.
***North American travelers (and all tourists) are required to purchase a US$10 tourist card ("Tarjeta de Turista") at the airport as soon as you enter the immigration area and before getting in line for immigration. It's a tax charged to visitors. To lessen your wait, have the exact US$10 bill ready and purchase the card from one of the automated machines, rather than standing in line at the ticket counter for change. You can also pre-purchase and print the card online at this link. Travelers from the following countries are exempt from the tourist card: Argentina, Chile, South Korea, Ecuador, Israel, Japan, Peru and Uruguay. UPDATE AS OF APRIL 2018: Tourists cards will be incorporated into all air and sea fares. Check out this blog post for more information.
Tourist cards are valid for 30 days. Should you overstay by a few weeks or months, you will simply bear a scaled exit charge at the airport when you leave. The current charges range from US$60 (RD$2,500) for 1 to 3 months overstay to US$115 (or RD$5,000) for one year.
The customs lines can be a pain or it can be easy, depending on the airport in which you land. Santo Domingo has the longest lines, but they usually don’t go through your things if you are traveling light and are a short-term tourist.
If you are not from North America or Europe, consult your Embassy to determine if you need to obtain a visa for entry to the DR.
For Americans staying a week or more, it's always a good idea to register your presence in the DR with State Department's Smart Traveler Enrollment Program (STEP)–for keeping up to date on any important travel alerts and news.
Excerpt from Moon Dominican Republic (Moon Travel Guides, 2016 edition)