An asylee is a person who meets the definition of "refugee" (see the box entitled What is a Refugee? on this page) who is already present in the United States, or seeking admission at a port of entry. A refugee who is already present in the U.S. or seeking admission at a port of entry may apply for asylum, a special form of protection that allows the person to remain in the United States. 8 U.S.C. §1158.
For more information about the application process for asylum, see this Asylum page on the USCIS website.
8 U.S.C. §1101(a)(42)(A) defines refugee as any person outside that person's county of nationality who is unable or unwilling to return to that country because of persecution or a well founded fear of persecution on account of race, religion, nationality, membership in a particular social group, or political opinion.
For more information about the process to receive protected status as a refugee in the U.S., see this Refugees page on the USCIS website. An application for refugee status must be made from outside the United States.