The U.S. Supreme Court's 2013 decision overturning the section of DOMA that declared marriage to be valid only if it was between a man and a woman was a victory for many binational same-sex couples. At last, they could apply for green cards (U.S. lawful permanent residence), fiance visas, and other derivative visas and benefits, based on legally valid same-sex marriages.
That might have come as small comfort, however, to binational couples who had already struggled for years to remain together, and found no legal options to do so. In some cases, the foreign-born half of the couple chose to take the risk of remaining in the U.S. unlawfully, and eventually was caught and placed into removal proceedings. The degree to which DOMA can help such couples now depends on how far along the proceedings got, where the noncitizen is now, whether the couple is legally married, and the basis upon which the noncitizen was ordered removed (deported).
This article will provide a brief review of these issues and possibilities. However, this is both a complex and a changing area of law. Absolutely see an experienced immigration attorney for a full personal analysis of your situation and for help in preparing the necessary legal paperwork.
The best scenario (though by now unlikely) is one in which the foreign national is still in the United States waiting for a decision from the Board of Immigration Appeals (BIA) or higher court and you are already married, or the immigrant is not in detention and you can marry. (So far, the immigration authorities have shown no inclination to let people out of detention in order to get married, but this could change.)
With a case that's still on appeal, you might then be able to file what's called a motion to "remand" it back to the immigration court. You would then present your application for a marriage-based green card to the immigration judge. See In Removal Proceedings: Who Can Apply for Adjustment of Status Based on Family? for information on eligibility and procedures.
Issues could arise, however, if the basis for the noncitizen's entry into immigration court proceedings or the court's negative decision also indicates that the foreign national is inadmissible to the United States. This could be due, for example, to conviction for a crime or the noncitizen's extended unlawful presence. That's part of why you'll want to get an attorney's help with this.
If you have a final order of removal on your record, your case is considered closed. If you remained in the U.S. regardless, you are considered a fugitive and can be arrested and removed at any time.
Talk to an immigration attorney about whether you might ask to "reopen" your case. This is a legal possibility in situations where new evidence has arisen, or where you are applying for a form of relief that was unavailable at the time of the earlier proceedings (in this case, a green card based on a same-sex marriage).
The possibility that you are "inadmissible" to the U.S. may, again, be a complicating factor. The very fact of your removal order makes you inadmissible for five, ten, or 20 years, or even permanently, depending on the reason for removal and whether you were convicted of a crime. The underlying reason for your removal may also give rise to separate grounds of inadmissibility. Nevertheless, it's possible to apply for waivers of both the underlying grounds of inadmissibility and the prior deportation. For more information, see After Removal: Possibilities for Reentry to the U.S.
A further complication might arise if you haven't yet married and the noncitizen is overseas. If it is legal in the non-citizen's country or the two of you can travel to a nearby country, great, as long as the U.S. citizen (if living in the U.S.) is willing and able to travel.
If not, the U.S. will not likely confer immigration benefits or admit the noncitizen to the U.S. based on a relationship that has not been legally recognized. Although a K-1 fiance visa is available to same-sex fiances of U.S. citizens, obtaining one might not be possible if you'll be applying for one of the inadmissibility waivers that requires a qualifying relative in the U.S., most likely a spouse. Again, however, you'll want to talk over strategy with a qualified immigration attorney.
For more information on same-sex immigration rights, see the "LGBTQ Rights and U.S. Immigration Law" page of Nolo's website.