An asylee is a person who has been granted asylum. Asylees are authorized to work in the United States, may apply for a social security card, may request permission to travel overseas, and can petition to bring family members to the United States. Asylees may also be eligible for federal or Office of Refugee Resettlement benefits, such as Medicaid or Refugee Medical Assistance.
After one year, an asylee may apply for lawful permanent resident status (i.e., a green card). Once the individual becomes a permanent resident, he or she must wait four years to apply for citizenship.
If you are granted asylum, please visit this page from USCIS on information on the Benefits and Responsibilities of Asylees in the United States.