If your fiancé falls under one of these circumstances and their immigration application has been denied, they can apply for an I-601 Application for Waiver of Grounds of Inadmissibility.
If you have been unlawfully present in the United States, you may be subject to either a three-year time bar or a ten-year time bar. If you are subject to a time bar, you will need to apply for a waiver.
The U.S. Department of State has indicated that the most common elements of a crime involving moral turpitude are (1) fraud; (2) larceny; and (3) intent to harm a person or property.
With such a variety of reasons to be found inadmissible it is not uncommon. Being found inadmissible is not a death sentence to your immigration process; however, some grounds of inadmissibility are more difficult to overcome than others.
In order to get a green card for family reasons, you and your fiancé will need to tie the knot and make it official. While you can get a fiancé visa prior to being married, you can only apply for a green card after your marriage.