An asylum-seeker is a person who claims refugee status but whose claim has not yet been processed.
The definition is inscribed in international law and is not subject to interpretation. Do not use migrant as a general term for refugee or asylum-seeker.
In times of mass migration, such as the current civil war in Syria, the United Nations refugee agency cannot reasonably process every refugee claim and thus considers people fleeing the situation prima facie refugees until their claims are otherwise refuted. Thus, a person fleeing the war in Syria should be referred to generally as a refugee unless otherwise legally-designated.
Refugee claims are processed by the U.N. High Commissioner for Refugees.