First, let’s get one important thing straight: Marriage to a U.S. citizen makes you eligible for a green card. Having a green card for a certain number of years can make you eligible for U.S. citizenship. But it’s a two-step process—at a minimum. In other words, even if your U.S. citizen husband sponsors you, you cannot become a U.S. citizen right away.

Below are the steps you will probably have to go through, and the typical amount of time each one will take.

(A note to anyone who isn’t in the exact same situation: Procedures differ for applicants who are living outside the U.S., or who are inside the U.S. but not in lawful immigration status, or who are marrying a U.S. permanent resident rather than a citizen. See this chart for the procedural scenarios.)

• Your U.S. citizen husband prepares a visa petition for you on Form I-130, issued by U.S. Citizenship and Immigration Services (USCIS) and you prepare an adjustment of status application (with Form I-485 and accompanying forms and documents) and you mail this package to USCIS.
• Several months later, USCIS calls you and your husband in for an interview. You are (if all goes well) approved for lawful conditional resident status. This is not yet permanent resident status, because you presumably will not have been married for a full two years by the time your interview date arrives.
• You are a conditional resident for two years. Ninety days before those two years are up, you and your husband submit a joint petition on Form I-751, asking that the conditions on your residence be lifted, thus allowing you permanent residence. Several months later, USCIS approves your request.
• If you remain married to and living with your U.S. husband for three years from the date of your approval for conditional residence, you can apply for U.S. citizenship (naturalization) by submitting Form N-400. This is an important exception: Most people must wait five years after getting a green card to apply for U.S. citizenship. But you need to have met all of the conditions for the entire three years, namely that you were living with your spouse the whole time, and that he was a U.S. citizen that whole time. (That means that someone whose spouse became a U.S. citizen after the immigrant got U.S. residence would have to wait three years from that point to take advantage of the exception, with the result that it might not even help them.)

If the marriage breaks up or you stop living together, you must count a full five years from the date of your approval for residence to apply to naturalize. And if your marriage runs into trouble or you get divorced before you’re able to get the conditions on your permanent residence lifted, you could be facing deportation as well. See What If Your U.S. Spouse Won't Sign the Joint Petition (I-751)?)

The news is also bad if the marriage ends due to the death of the U.S. spouse, as described in My U.S. Citizen Spouse Died: Can I Still Apply to Naturalize After Three Years?.

For comprehensive information on the green card application process, see Fiance and Marriage Visas: A Couple’s Guide to U.S. Immigration. Or, if you have any questions about complicating factors in your case, or wish help with the application process, consult an immigration attorney.