After you have been a permanent resident (i.e. received your green card) in the US for five years, you are eligible to apply for US citizenship.

The application process is:

• Fill out the N-400 form. Mail the form to the processing center that covers your state, together with two passport photos, a check to cover the processing and any other fee as instructed, and a photocopy of both sides of your green card. If husband and wife are filing together, put both applications in the same envelope so that your finger printing and interview will be scheduled for the same time.
  If you have taken a trip outside the US for more than six months, then you need to provide evidence showing your tie to the US. Such evidence can be your tax return form, your US pay stub, your US mortgage or rent payment record.
• Get fingerprinted. You will receive appointment letter for the time and location to get your fingerprints taken. After this, you can look up this web page to check the processing status of your N400 application, among other applications.
• Take the interview. You need to bring your ID including your passport and green card. You will be asked questions about your application and background. Then you will take the English and civics test.
• Take the oath. If you have passed the interview, you will receive a notice to attend a Naturalization Oath Ceremony. On the back of the notice, you will be asked about what you have done since your interview, such as if you have traveled outside the US. After taking the oath, you will receive your Certificate of Naturalization. This is proof that you are a US citizen. You are recommended to apply for a US passport soon after this ceremony.