A foreign-born person who intends to marry a U.S. citizen might be eligible to come to the U.S. for the wedding, using a K-1 or "fiancé" visa. (The law that covers this is Immigration and Nationality Act (I.N.A.) § 214 or 8 U.S.C. § 1184, with regulations at 8 C.F.R. § 214.2(k).)
The K-1 is a nonimmigrant (temporary) visa, which gives its holder six months in which to enter the U.S. (single-use) and then allows a stay in the U.S. of only 90 days. After you get married, however, you can apply for a green card (become a permanent resident) in the U.S., through a process called "adjustment of status."
Fiancés are fortunate in that there are no annual limits on K-1 visas, and therefore no long waiting periods such as occur in some other visa categories.
If you are the would-be immigrant and are already living in the United States, or if your U.S. citizen fiancé lives outside of the U.S. with you, getting a K-1 visa is probably unnecessary. Instead, you might be better off getting married and then applying for a green card. (Consult an immigration attorney for the details and a personal analysis.)
The main eligibility criteria for getting a K-1 visa are that:
The K-1 visa is not appropriate for couples who are merely thinking about getting married. You'll need to show proof that you truly plan to get married, such as letters to each other discussing your plans, and wedding announcements sent to friends.
One of the most convincing types of evidence of your intentions is documents showing that you've actually set a date for the wedding and made some arrangements like renting a space and hiring a photographer. (But leave room for flexibility—you can't necessarily count on getting a visa in time for your planned date, and it would be terrible to have to cancel and lose your deposits.)
For some couples, the requirement that they have already met is difficult or violates their religious principles. If you practice a religion in which marriages are customarily arranged by families and premarital meetings are prohibited, you can ask that U.S. Citizenship and Immigration Services (USCIS) waive the personal meeting requirement. You'll have to show that both parties will be following all the customs of marriage and weddings that are part of the religion.
It is also possible to obtain a waiver of the personal meeting requirement if such a meeting would cause extreme hardship to the U.S. citizen half of the couple. Only the most extreme situations, usually involving medical problems, are likely to be regarded as sufficient reason for the waiver to be granted. Economic problems alone are not usually acceptable. So if it's at all possible for the two of you to meet in person, you should do so.
When you get a K-1 visa, any of your unmarried children under the age of 21 can be issued K-2 visas. This will enable them to accompany you to the U.S. They, too, will be able to apply for green cards once you get married, even if they've since turned 21 (after U.S. entry).
For more on the process you'll need to complete, see Steps to Getting a K-1 Fiance Visa.