The unmarried children of a U.S. citizen, so long as they are under the age of 21, are potentially eligible for U.S. green cards (lawful permanent residence). They are what’s called "immediate relatives," meaning that they face no annual limits on the number of visas (green cards) given out in their category. They can receive a green card just as soon as they can get through the application process.
Stepparents and stepchildren qualify as immediate relatives who can petition for children if the marriage creating the parent/child relationship took place before the child’s 18th birthday. So, for example, if you are a U.S. citizen who married a foreign-born person, and you are applying for a green card for your spouse, you could, so long as you got married before the children turned 18, apply for green cards for them at the same time. (It doesn’t matter if they turn 18 before the application process is done.) But they will still need you to file separate forms and paperwork on their behalf.
Another thing worth checking on: If you were a U.S. citizen when your child was born, your child might have already become a citizen, automatically.
Be aware that the married children of a U.S. citizen, or any children over 21, are considered "preference relatives." Annual limits apply to the availability of visas in their categories, resulting in waits of a few years, within the middle of the application process, until a visa becomes available. Learn more about this inevitable wait in How to Determine Your Priority Date for Immigration Purposes.
If you are the U.S. citizen parent wishing to sponsor a family member to immigrate, it will be up to you to take the first step in the process. This is to prepare and file Form I-130, issued by U.S. Citizenship and Immigration Services (USCIS).
It’s important to realize that every child of a U.S. citizen needs a separate I-130 filled out for them. You can’t combine children on one form, and you can’t (if you’re also petitioning for a spouse), list them as children and expect them to be included automatically as "derivatives." Through a quirk in the law, petitioners with green cards who apply for their spouse and children can do it all with one I-130, but you, as a U.S. citizen petitioner, cannot.
You will need to pay a nonrefundable fee along with Form I-130.
If your child already lives in the United States, then they might be eligible to adjust status—that is, apply for and receive a green card without having to leave and be interviewed at an overseas U.S. consulate. This is probably possible only if your child is either living in the United States legally with a visa or some other status, or entered with a visa or after ESTA registration using the Visa Waiver Program or VWP (even if the permitted stay has expired).
For more information on the forms, documents, and fees required, see What You Need to File I-485 for Adjustment of Status.
Better yet, if your child is eligible to adjust status in the United States, you don’t have to wait for USCIS to issue an I-130 approval to apply for adjustment. Simply include the completed Form I-130 (with fee) in your adjustment of status packet when you submit the entire packet to USCIS. If you're not sure about eligibility, have a lawyer help at this stage.
If your child lives outside the United States, then they will need to complete the process, including attending an interview, at a U.S. embassy or consulate in their country of residence (or, if you request it of the NVC, their country of nationality).
For more information on the forms, documents, and fees required for this, see What Happens During Consular Processing?
At the U.S. consulate, the child should be given an immigrant visa, allowing entry into the United States. It’s important to understand that your child is not a permanent resident yet. That happens only when the child enters the United States using the immigrant visa. For that reason, it’s very important that the child not get married before entering the United States, as described in Why Children Must Remain Unmarried Until Green Card Approval.
If you're looking to get U.S. lawful permanent residence for your child or children, you might want to talk to an attorney. Hiring one to evaluate the child's eligibility and assist with the paperwork and petition process adds legal fees, but a professional may help to expedite your case and avoid any legal problems.