If you are a foreign national who has been approved for asylum in the United States (an "asylee"), you can apply for adjustment of status to permanent resident (otherwise known as getting a green card) one year after the asylum grant. (That assumes you have also been physically present in the United States for one year's time; if not, you'll have to wait longer.)
You can also wait longer than a year to apply, if you wish to, or if you need that time in order to meet the one-year's 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) could decide that you no longer need the protection of the United States. It could not only deny the green card but place you into removal proceedings for possible deportation to your home country.
The primary application for the purpose of applying for a green card while in the United States ("adjusting status") is USCIS Form I-485, Application to Register Permanent Residence or Adjust Status. However, the I-485 is just the first item in a packet of forms and documents you will need to submit, which should also include:
In addition, you will need to pay a fee for the application and biometrics (fingerprinting), unless you qualify for a fee waiver. The latest fee amount can be found on the USCIS fee schedule, and further instructions can be found at on the I-485 page of www.uscis.gov.
Fortunately, as an asylee, you are exempt from the public charge ground of inadmissibility. That means you do not need to prove that your income takes you above the poverty line in the United States. Thus if your income is in fact low, it might be worth applying for a waiver of the fee, using USCIS Form I-912.
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 the associated fees with either using a credit card (in which case you must also file Form G-1450) or by making an electronic transfer from a U.S. bank (using Form G-1650, Authorization for ACH Transactions). Owing to a change in USCIS policy, checks, bank drafts, and money orders are no longer accepted.
USCIS will inform you the time and location of your appointment for biometrics (collecting fingerprints, a signature, and so forth).
If your adjustment application is deemed incomplete, USCIS could return the whole packet to you for refiling; or it might follow up with a Request for Evidence (RFE). The RFE will give you a deadline within which to submit whatever USCIS is asking for. Don't miss the deadline, or your case will likely be denied! If you aren't able to submit all that's asked for within the allotted time, send what you have and explain the situation. Also include any documentation showing why a certain item is impossible to obtain.
Next, USCIS will send you an appointment date for a personal interview at a USCIS office. There, an officer will review your Form I-485 and other materials, ask you questions, and finally make a decision on your adjustment of status application. (However, USCIS might ask for further documents or follow-up instead.)
If you are approved for lawful permanent residency, a green card will be sent to you by mail some weeks later. At that time, you will be on your way to qualifying for naturalized U.S. citizenship. And, there's good news: one year of your time as an asylee counts as permanent residence in getting to the five years of lawful permanent residence you'll need to quality for U.S. citizenship. (This is a concept called "rollback." If you look at your green card, you should see that your approval date has already been back-dated by one year to reflect this.)
If you want to apply for adjustment of status as an asylee, you might wish to consult with an experienced U.S. immigration attorney. That's especially true 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 immigration application process and accompany you to your adjustment interview at a USCIS office. See, for example, How to Find an Excellent Immigration Lawyer Near Me.