Formally, the Office of the United Nations High Commissioner for Refugees; office may be omitted.
Colloquially, the United Nations refugee agency, acceptable on first mention with the formal name written elsewhere in the story; refugee agency is not the official name and should not be capitalized.
Abbreviated as UNHCR, acceptable on second mention after the formal name:
The Syrian refugee crisis is worsening, the United Nations refugee agency said Saturday. According to the U.N. High Commissioner for Refugees, one million people reached Europe by sea in 2015. Nearly one in 20 people worldwide is a refugee, the UNHCR said.
The head of the UNHCR is the High Commissioner for Refugees. Their name should proceed the title: Filippo Grandi, United Nations High Commissioner for Refugees.
Do not confuse with UNCHR, a one character transposition. This is the United Nations Commission on Human Rights, which was replaced in 2006 by the United Nations Human Rights Council (UNHRC).
See United Nations.