A US Citizen can apply for a special temporary visa to allow a non-citizen (their fiancée) to enter the country in order to get married to a US citizen inside the US.
Note: If the non-citizen is already a legal resident in the US, or if you plan to marry outside of the US, you do not need a K1 visa. A person visiting on a tourist visa is NOT a legal resident.
Marriage To A Non US Citizen with a K1 Visa or a Tourist Visa?
Some individuals incorrectly believe that a K1 visa can easily be replaced with a standard tourist visa. However, USCIS generally will make the immigration progress more difficult for a non-citizen who comes to the US on a tourist visa and marries during their stay. USCIS will assume that this was done to circumvent the visa process and that could be seen as a violation of the law. It is not recommended that couples marry on a tourist visa in the US, but instead file I-129F for a K1 Visa prior to getting married in the US.