If you are an asylee, you can apply for adjustment of status to permanent resident -- otherwise known as getting a green card -- one year after the grant of asylum, assuming you have also been physically present in the United States for one year's time.
You can also wait longer than that to apply, if you wish, or if you need to in order to meet the physical presence requirement. But the longer you wait, the higher the risk that conditions in your home country will normalize -- in which case U.S. Citizenship and Immigration Services (USCIS) may decide that you no longer need the protection of the United States and not only deny the green card but place you in removal proceedings.
The primary application for this purpose is USCIS Form I-485, Application to Register Permanent Residence or Adjust Status. However, this is just the first of a packet of forms and documents you'll need to submit, which include:
In addition, you'll need to pay a fee for the application and biometrics (fingerprinting). The latest fee amount and further instructions can be found at www.uscis.gov.
For a more comprehensive discussion of the forms, see What You Need to File I-485 for Adjustment of Status.
The completed Form I-485 must be filed with the USCIS office having jurisdiction over your place of residence in the United States. You can pay both any and all fees with a single check.
USCIS will inform you the time and location of your fingerprinting. If your application is incomplete, USCIS will return the whole thing to you for refiling. USCIS will next call you for an interview. There, a decision will likely be made on your application (though USCIS may ask for further documents or follow-up instead).
If you want to apply for adjustment of status as an asylee, you may wish to consult with an experienced U.S. immigration attorney for help -- especially if your country has been in the news lately, with word of a civil war or conflict having ended or conditions having otherwise improved. The attorney can assist you with the entire process and accompany you to your adjustment interview.