A migrant is: "A person who travels to another country or place, often in search of work." (Cambridge University Dictionary)
An Asylum Seeker is: Anyone in the world who has "a well founded fear of being persecuted for reasons of race, religion, nationality, membership of a particular social group or political opinion" (1951 UN Convention on Refugees).
This would include Civil war: Syria, Sudan, Afghanistan; Religious persecution: Christians in Iran, Iraq, Ahmadi Muslims and Bahai's in Pakistan Sexuality: people who identify as LGBT in Uganda, Kenya, Pakistan, India. Ethnic Tamils in Sri Lanka.
The UK asylum system offers this protection but is strictly controlled and complex. The burden is on the asylum seeker to prove that their claim is genuine.
A refugee is: A person who has applied for asylum and who has had that claim accepted. Such persons will be granted "leave to remain", usually for a five year period, after which if the reasons that led them to claim asylum persist, they can apply for "indefinite leave to remain: and eventually they can apply for citizenship eg. Mo Farah.