If you receive a K-1 fiancé visa with which to enter the U.S. and get married to a U.S. citizen, your unmarried children under the age of 21, whether or not they are the biological children of your U.S. citizen fiancé, might be eligible to accompany you. They will be given "K-2" visas for U.S. entry. The term "children" includes not only natural children, but adopted children and any born out of wedlock, if your home country legally recognizes them as yours.
Don't be confused by the fact that already-married applicants for U.S. green cards need to prove that their children fit the definition of "stepchildren," by showing that the parents' marriage took place before the children turned 18. You, as a fiancé visa applicant, don't need to fit those criteria. The only reason we even mention it is that immigration officials themselves sometimes get confused about it and try to deny K-2 fiancé visas to children who are over 18 but still under age 21. If this happens to you, understand that the official has gotten the laws mixed up, and point to the U.S. immigration regulations at 8 C.F.R. § 214.2(k)(3).
Your children will have to go through the same (or a very similar) visa application process as you. At the beginning of the application process, all the U.S. citizen sponsor needs to do is to include the children's names in Part 2 of the Petition for Alien Fiancé (Form I-129F) that they file with U.S. Citizenship and Immigration Services (USCIS).
The consular officials that will ultimately handle the visa case (after USCIS approves the I-129F and forwards the file) should send extra sets of the various required forms for the children to fill out. For young children, it's okay for a parent to fill out a form and even sign it on their behalf. (Just sign your name, then write Parent of [name of your child].)
The children will, like you (their K-1 applicant parent), have to prove that they are not inadmissible to the U.S. and that they will be financially supported along with the immigrating fiancé. Each child will need to show proof of their relationship to the immigrating K-1 parent, such as a birth or adoption certificate (possibly with English-language translation).
After you and the U.S. citizen petitioner get married and you apply for a green card (to "adjust status" to lawful permanent residence), your K-2 children may apply along with you. Each will need to submit (by mail) a separate application for adjustment of status to U.S. Citizenship and Immigration Services (USCIS).
For planning purposes, note that the children must remain unmarried and under age 21 right up to the day they enter the United States on their K-2 visas. Fortunately, if you alert U.S. immigration authorities to an upcoming 21st birthday, they can usually speed up the processing for you.
(Unfortunately, a law you might have heard of called the Child Status Protection Act does not protect children on fiancé visas from the loss of visa rights caused by their turning 21.)
Check your own country's law on removing your children if their other parent is staying behind. If you are planning to bring children to the United States who are not the biological children of your fiancé, it will be up to you to comply with your country's custody requirements. Even if the children are legally in your custody, you might need to get written consent from the other parent for you to take the children out of your country.