What happens when an American citizen wants to marry a person living in the US without proper papers?
Maybe they get a good lawyer and try to do whatever is required to attain legal residency.
But that process is risky, and one possible outcome is that the "alien" (who may have been living in the US since infancy) may be deported and required to spend ten years in a country he or she doesn't really know before then applying to emigrate to the US.
In some cases, the American goes with his/her fiance to live abroad until the would-be spouse can legally enter the US and they can legally marry (which then can lead to the spouse gaining citizenship).
One woman who knows this scenario intimately is Giselle Stern Hernandez. Look at her website and learn about her story.