You can travel to America without a visa if you are a citizen of a country comes under the Visa Waiver Program (VWP). The VWP is administered by the Department of Homeland Security of the United States and allows citizens of 38 nations to travel to the US without a visa, for tourism or business purposes, and stay up to 90 days.
Some of the countries that are part of this Visa Waiver Program are Hungary, Australia, Germany, Japan, Switzerland, Belgium, Italy and many more. You can see the list of other countries on the official website of the US Department of State, Bureau of Consular Affairs.
However, under this program, you are not allowed to extend the period of stay beyond 90 days. And, to travel to America under the VWP, you need to have an Electronic System for Travel Authorization or ESTA prior to boarding a US-bound carrier. For this, you must have a valid e-passport. And, you need a valid e-passport for each individual travelling to the US with you. For more details on the ESTA, you can check the US ESTA site.
In addition, if you are able to show proof of an urgent situation, such as a dying relative or friend, you might get permission to travel to the US without a visa for a limited period of time on humanitarian grounds. It is advisable to only state genuine reasons to get permission to travel to America without a visa. If you are found to have falsified information, the consequences could be severe.