Embassy of the United States Copenhagen, Denmark

If you're a Danish citizen, you probably don't need a visa to visit the U.S.  for tourism and/or business.  Instead, you can travel to the U.S. under the Visa Waiver Program (VWP).
To make your entry into the U.S easier, the Department of Homeland Security has developed the Electronic System for Travel Authorization (ESTA): an online application system that tells you if you need a visa or if you can travel under the VWP. 
If you can answer YES to all of the four questions below, you only need an ESTA authorization to travel to the United States:

Question 1:  Are you a citizen of Andorra, Australia, Austria, Belgium, Brunei, the Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Japan, Latvia, Liechtenstein, Lithuania, Luxembourg, Malta, Monaco, the Netherlands, New Zealand, Norway, Portugal, the Republic of Korea, San Marino, Singapore, the Slovak Republic, Slovenia, Spain, Sweden, Switzerland, or The United Kingdom?  If not, you need to apply for a visa.

Question 2: Is your trip to the United States for business or tourism purposes? If not, you need to apply for a visa.

Question 3: Will your stay in the United States (to include side trips to Canada or Mexico) be for 90 days or less? If not, you need to apply for a visa.

Question 4: Is your passport Machine Readable?  You will need an e-passport if your passport was issued after 26 October 2006, or if you are a citizen of the Czech Republic, Estonia, Latvia, Lithuania, Hungary, the Republic of Korea and the Slovak Republic.  An e-passport is a passport that contains a computer chip holding the same information printed on the passport’s data page.  For more information on e-passports and Machine Readable passports, please download the U.S. Customs and Border Protection's information sheet.  If not, you need to apply for a visa.

If you are eligible to use ESTA, please visit the ESTA web page at https://esta.cbp.dhs.gov. If you received an approved travel authorization after using ESTA, you may travel to the United States without a visa.

VWP travelers should be aware that unauthorized third-parties have established web sites which charge visitors seeking to travel to the United States under the VWP, for information about ESTA and for submitting ESTA applications to DHS on behalf of the traveler.  These businesses and web sites are not endorsed or associated with the United States Government.  Free information about ESTA can be found on the official ESTA web site.  However, individuals applying for and obtaining a travel authorization through the official ESTA web site will now be charged a fee as described on the web site.