If you marry a United States Citizen, you can begin the process immediately to get your Green Card. Depending on your current status and whether you entered the country legally, you can stay in the United States and Adjust your Status to that of a Lawful Permanent Resident. Otherwise, you may have to have the application prepared and sent to the United States Consulate of your home country.
If you have been married less than two years, you will get a Conditional Green Card. This means that two years later, you will have to file another petition to remove the Conditional Residency to prove the marriage is real.