When you are granted asylum, you do not automatically become a permanent resident.
But, when you are granted asylum, you will be able to lawfully remain in the United States as an asylee. You will also be able to get an Employment Authorization Document (EAD) or work card so that you can prove that you can lawfully work in the United States. If you need to travel outside of the United States, you’ll be able to apply for a Refugee Travel Document. Basically, you will be able to live your life in the United States and will be able to build a future for yourself and your family.
However, you will not be considered to be a lawful permanent resident until after your application for a green card is approved. This is a separate process than your asylum application.
The process that you will likely use to apply to become a permanent resident is the adjustment of status process. You can submit your adjustment of status application after you have been an asylee for at least one year. You should expect it to take at least four months for your application to be approved, and in some cases it could take over a year before your application is approved. When your application is approved, you will be sent a green card.
Before you submit your application for adjustment of status, you should make sure that you meet all of the requirements. If you have any questions about whether you qualify to become a permanent resident, you should discuss your case with an immigration lawyer before you submit your application.