If your fiance entered the United States on a K-1 Fiance Visa, but you for some reason didn't get married and apply for adjustment of status (a green card) within the allotted 90 days, then your fiancé has overstayed the visa. He or she is expected to leave the U.S. right away. If caught here, your fiance will most likely be placed into removal proceedings.
However, you wouldn't be the first person to face this situation, and most couples deal with it in a different manner than having the immigrant leave the United States. You can act quickly, get married, and file not only a Form I-130 visa petition (indicating that you are a U.S. citizen and would like to sponsor your new spouse for a green card), but also the remainder of the green card (adjustment of status) application.
IMPORTANT NOTE: We're assuming that you, the U.S. citizen, were the original petitioner on your spouse's fiance visa petition. If he or she got the K-1 visa with the idea of marrying someone else, but decided to marry you instead, the analysis below will not apply. Your fiance would need to leave the U.S. -- as soon as possible, to avoid immigration penalties for overstaying a visa by six months or more -- and apply for either a fiance or marriage-based visa from there.
You'll first need to not only get married, but to obtain the official, government-issued certificate that provides evidence of your marriage.
Next, you'll need to obtain Form I-130, Petition for Alien Relative, Form I-485, Application to Register Permanent Residence or Adjust Status, and various other supporting forms (such as the I-864 Affidavit of Support). They are all issued by United States Citizenship and Immigration Services (USCIS). You'll mail them all in one package to a central USCIS office.
After submitting this package, you can rest easy. Your spouse is legally in the United States. Assuming your spouse also submitted Form I-765 with the packet, a work permit card (EAD) should arrive some weeks after you submitted the packet. That's meant to cover the time up until your adjustment of status interview at a USCIS office, at which your spouse's green card should be approved.
Another thing that will happen before the green card interview is that your spouse will be called in for "biometrics," or fingerprinting. That allows for a criminal and immigration record check.
For details, our article on the green card process after the marriage.
If your fiance gets picked up by the immigration authorities and placed into proceedings before you can implement this strategy, you can still get married and submit your I-130 and other paperwork directly to the judge. However, it will receive tougher scrutiny than if you had simply handed everything in without the pressure of a court proceeding. And, of course, you'll have to pay higher fees to the lawyer who you'll need to represent you in court.
If your fiancé has a visa overstay of fewer than six months, he or she could also leave the United States and apply at a U.S. consulate to return to the United States on another fiancé visa. However, the law prohibits applying for more than one K-1 fiance visa in the course of two years without obtaining a waiver. Your fiance would have to do a lot of explaining as to why he or she didn't comply with the terms of the first visa.
If your fiancé has a visa overstay of six months or more, leaving the United States is definitely not a good idea. He or she could be barred from reentering the U.S. for three years (if the overstay was between 180 days and one year) or ten years (if the overstay was over one year). For more on this issue, see "Three-Year and Ten-Year Time Bars for Unlawful U.S. Presenc."