If you are an American citizen, you can visit the Dominican Republic with either a passport and visa or a passport and tourist card. You must have (with your passport) either a visa or a tourist card. If you have a visa, you will not need a tourist card.
According to the Embassy of the Dominican Republic in the United States:
1) Does an American citizen require a Visa to enter the Dominican Republic as a tourist?
No, American citizens do not require a visa to enter the Dominican Republic as tourists. You just need to purchase a tourist card (Fee US$10.00) online... or at your port of entry in the Dominican Republic.
2) Is a visa required for non US citizens to enter the Dominican Republic as a tourist?
Any person who can legally travel or reside in the United States, Canada and the European Union (including Great Britain) does not need a Visa to visit the Dominican Republic for tourist purposes and can enter the Dominican Republic with a Tourist Card and a valid passport.
According to the United States Department of State, under Entry, Exit, and Visa Requirements, "Visitors who do not obtain a Dominican visa must purchase a tourist card at the airport for 10 USD upon entry or online prior to travel." So, you will only need to purchase a tourist card (which allows you to stay up to 30 days) if you have not obtained a visa.
Please refer to the links below, which also includes phone numbers for the United States embassy in Santo Domingo, Dominican Republic.