Let’s start with what a refugee is not: a refugee is not an illegal alien or someone leaving their country for economic opportunities elsewhere. The UN Convention Relating to the Status of Refugees defines a refugee as a person who “owing to a well-founded fear of being persecuted for reasons of race, religion, nationality, membership of a particular social group or political opinion, is outside the country of his nationality, and is unable to or, owing to such fear, is unwilling to avail himself of the protection of that country or return there because there is a fear of persecution…”
Asylum seekers are individuals seeking refugee status in another country. International law recognizes the right of persons to seek asylum but no country is required to provide asylum. Most countries, including the U.S., require asylum seekers to undergo a legal process to determine if the person fits the definition of being a refugee. Countries may also require security clearances to ensure the refugee presents no danger to its population. The U.S. has an extensive security clearance process that is even more involved for refugees from Syria.
See this University of Minnesota fact sheet for much more on what constitutes refugee status and discussion of various international agreements that govern the rights of refugees.