If you are in the U.S. with an F-1 student visa and have married a citizen of the United States, you are eligible to file for a U.S. green card. The procedure for doing this is called Adjustment of Status (AOS). Using this procedure, you can file all your paperwork with U.S. Citizenship and Immigration Services (USCIS) and attend your interview in the United States.
Under the law, you're expected to file the appropriate forms with USCIS before your permitted stay under your F-1 student visa expires, which is usually 60 days after finishing the degree or any authorized period of Optional Practical Training (OPT). (See Can You Stay in the U.S. Legally While in Between Visas?)
Violating the terms of your F-1 visa, for example by working off campus without authorization, can also render it void, in which case you'd be expected to leave the U.S. right away.
Even if your permitted stay has expired, the fact that you entered the United States legally and are marrying a U.S. citizen allows you to apply for adjustment of status. The overstay becomes almost irrelevant (unless the immigration authorities happen to arrest you before you submit your green card application, in which case you'd be placed into removal proceedings).
If more than six months have passed since you were in valid F-1 status, your best bet might be to apply to adjust status. If you use the alternate procedure, in which you leave the U.S. and process your application at a U.S. consulate in your home country, you risk being hit with a penalty for your time spent in the U.S. unlawfully; that is, a bar on returning to the U.S. for three years if you overstayed by 180 days or more; or, if your unlawful presence lasted 365 days or more, the bar would be ten years.
(It's a lower risk for students than others, since USCIS or an immigration judge would have had to rule that you were in the U.S. unlawfully to trigger the counting of your unlawful time, but we wanted to alert you to this risk.)
See a lawyer for a complete personal analysis if you have any question about your current immigration status in the United States.
U.S. immigration authorities will accept a marriage that's conducted overseas; but if you have overstayed and need to adjust status, it's safer not to leave the United States for your wedding.
The steps to marrying a U.S. citizen within the United States depend on your local jurisdiction. The usual restrictions on marriage concern your respective ages, whether you are blood relations, and of course whether either of you is already married to someone else. Contact the local county courthouse to inquire. In some states, you can simply present yourself at the courthouse and get married that day. In others, you must prove local residency, take a blood test, and more.
You will need a government-issued copy of your marriage certificate in order to apply for your green card. The certificate that a church or temple might give you is not enough.
Applying to adjust status in the United States basically involves submitting forms, copies of documents, the results of a medical exam on the immigrant, and the appropriate fees, and later attending an interview at a USCIS office.
The main forms you will need to submit include USCIS Form I-130, Petition for Alien Relative, which the U.S. citizen fills out; and USCIS Form I-485, Application to Register or Adjust Status, which the immigrant fills out. Another important form is the I-765, Application for Employment Authorization, which allows the immigrant to obtain a work permit. And, the U.S. citizen will need to promise financial support, and show the ability to support the immigrant, on Form I-864.
These and the other needed forms are available for free download on the USCIS website at www.uscis.gov. Your AOS packet must be submitted by mail, not in person.
Note: While most types of immigrants must submit form I-130 separately, and wait for USCIS to answer before proceeding with the green card application, the fact of your legal entry and marriage to a U.S. citizen allows you to submit the entire packet to USCIS at one time (concurrently).
Some weeks or months after you submit the paperwork to USCIS, the immigrant will be called in for biometrics (fingerprinting). Weeks after that, the two of you will be called in for a personal interview at the USCIS Office.
At the interview, a USCIS official will review the paperwork and make sure the immigrant is not inadmissible (for example, hasn't committed any crimes or been diagnosed with a communicable disease of public health significance (during the required immigration medical exam), and doesn't appear likely to become a public charge because the U.S. citizen cannot provide sufficient financial support).
In addition, because this is a marriage-based case, the official will ask both of you personal questions to make sure that your marriage is the real thing, not just a sham to get a green card.
Because the paperwork is extensive and complicated, you may wish to have a lawyer help you. That's especially if there's any chance that the immigrant violated the terms of the student visa or is inadmissible. Or, if your case is relatively straightforward, consult the book Fiance and Marriage Visas: A Couple's Guide to U.S. Immigration, by Ilona Bray (Nolo).