You can stay depending on what you qualify for. If you have no close relatives in the United States who can sponsor you, you may be able to apply for another type of visa. If you have money to invest, you may be able to apply for one of the different types of business visas described in more detail in the Business section of this website. If you want to further your education or even if you want to learn English better you may apply for a student visa. IMPORTANT: You must apply to change your visa before your stay expires, which is usually six months but sometimes less. If you are escaping danger in your country and you have been threatened or attacked or arrested because of your religion or political opinion, etc. you may file for asylum. You must apply for asylum within one year of entering the United States.
Unfortunately, once you status expires, you are out of status, your options are much more limited. You cannot apply for any visa once you are out of status. You can apply for asylum, but if it has been more than one year since your entry into the United States, you must give a very good reason why you filed late such as changes in your country condition. Not knowing about asylum is not a good enough reason.
If you file for adjustment of status, in my recent experience it takes anywhere from about six months to about eighteen months to get to an interview. If all goes well at the interview you should get your green card within a few weeks. Sometimes, if there are complications or if the officers needs to review the file, it can several months. Generally, immigration officers tel you to give sixty days for a green card.
Unfortunately, there is little that can be done effectively. Sometimes green card or citizenship cases can drag on for months or even years. Sometimes, the officer handling the case changes, they are are waiting for a file to transfer or they are doing extensive background checks.
The best recourse is to call USCIS at 800-375-5283. You can speak to an officer who will take your information and send you what is called a "service request" in two to four weeks. Unfortunately, the officer on the phone cannot give much information and the letter will tell you not much more except that it is pending. They will never answer how much longer it will take. Sometimes, if that does not help you can request what is called an "Infopass". That is an appointment to go to the local immigration office so that an officer can look into your case and hopefully give more information.