If you have married a U.S. citizen or permanent resident, and applied for U.S. lawful permanent residence yourself (a green card) on that basis, an annulment is likely to either complicate or cancel your application process. Much like a divorce, the exact effect will depend on how far along in the application process you are. We'll discuss each of the possibilities below.
If your U.S. spouse has started the process for you by filing a petition on Form I-130 with U.S. Citizenship and Immigration Services (USCIS), and even if that petition has been approved, you yourself have no rights to enter or remain in the U.S. on that basis, much less to apply for a green card via that marriage.
In theory, you still have many application steps ahead of you, in particular filing your own application materials with the U.S. government and attending an interview at a USCIS office. In the course of this, you'd be expected to prove that the marriage is ongoing and that your U.S. spouse continues to be willing and able to sponsor you financially (or more literally, pay back the government if you end up needing public assistance).
If the interview is held in the United States, your U.S. husband or wife will be expected to attend. If it's is held at an overseas consulate, spouses aren't normally required to be there, but the marriage must nevertheless be ongoing.
The purpose of family-based green cards is to promote family unity, and after an annulment, the immigration laws figure there's no family in such a case to unify.
Unless you had already been married for two years or more when you applied for the U.S. green card, you will have received conditional residence status (which is essentially a green card, but one that expires in two years unless you prove that you deserve a permanent green card).
If you did receive conditional residence, it's possible to go forward toward a permanent green card even after an annulment of your marriage. However, you'll need to request a waiver of the usual requirement that you and your U.S. spouse jointly file the petition asking to convert from conditional to permanent residence (Form I-751).
To request such a waiver, you'll need to prove that:
It's crucial here that you provide documents from the last two years showing that your marital relationship was, in fact, the real thing, not a fraud. The same kinds of documents that you've shown the immigration authorities before will help: evidence of having bought or rented a house together, of having combined your finances with joint bank accounts and credit cards, photos from vacations together, and so forth, will all be helpful. But make sure they're more recent than the documents you presented with your green card application.
Realize also that you're not trying to prove you had a good marriage! Even letters from marriage counselors and the like can help prove that it was the real thing. Struggling with marital issues is something that many real couples go through.
You'll also need a copy of your annulment certificate. If the annulment isn't yet final by the time you need to file Form I-751 (which is due within the 90 days before the expiration of your conditional resident status), that can be problematic. Talk to an attorney. Whatever you do, get the Form I-751 turned in to USCIS on time, even if you don't have the certificate yet. (You'll probably then receive a Request for Evidence or RFE, with a deadline some weeks in the future.)
See Documents Needed for Filing an I-751 Form for more on what you're trying to show, and how to show it.
If the annulment isn't yet final, you might also look for alternative grounds upon which to request a waiver, such as if you were the victim of battery or extreme cruelty by your U.S. spouse, or would face extreme hardship if removed from the United States.
If you've already received U.S. permanent resident (a green card that doesn't expire in two years), an annulment should not affect you. You have a legal right to retain that status, assuming the original marriage was not a fraud to obtain a green card.
However, if and when you apply for U.S. citizenship, USCIS will have another opportunity to look into whether your marriage was a real thing. You can expect the annulment to raise questions, though nothing is automatic at this point. You should collect and save any documents showing your attempts to make a life together, which you'll likely be asked to present to USCIS at or following your citizenship interview.