Born in Winnipeg, Manitoba, Carr is a descendent of Jewish immigrants who arrived from Russia in 1906. He had his bar mitzvah at Shaarey Zedek Synagogue (Winnipeg) in 1964 and was educated at the University of Manitoba and McGill University. He worked as a journalist, musician and administrator, serving as both an oboist and trustee with the Winnipeg Symphony Orchestra on different occasions. Before entering public life, he was also executive director of the Manitoba Arts Council.
On June 3, 1969, Carr and four other Young Liberals joined the New Democratic Party of Manitoba so as to support Sidney Green's unsuccessful bid to lead the party. He later returned to the Liberals. During the early to mid 1990's, he worked as a reporter for the Winnipeg Free Press.
Carr first ran for the Manitoba legislature in the provincial election of 1986, losing to Progressive Conservative leader Gary Filmon in the upscale west Winnipeg riding of Tuxedo. He ran again in the provincial election of 1988 during a period of increased support for the provincial Liberals, and scored a surprisingly strong victory in the riding of Fort Rouge, defeating NDP cabinet minister Roland Penner by over 2000 votes. The Progressive Conservatives formed a minority government after this election, and Carr became Deputy Leader of the official opposition party.
Following redistribution, Carr sought re-election in the 1990 provincial election in the riding of Crescentwood, where he defeated Progressive Conservative Tom DeNardi by 1310 votes. The Liberals fell from twenty seats to seven in this election, and Carr resigned his seat in February 1992.
From 1992 to 1997, Carr was on the editorial board of the Winnipeg Free Press. In 1998, he became president and CEO of the Business Council of Manitoba, an organization which he co-founded. Carr is also a board member on the Canada West Foundation, and the Winnipeg Airports Authority, and well as vice-president of the Performing Arts Consortium of Winnipeg, and a director of the Manitoba Arts Stabilization Fund. Carr, who is Jewish, has also been active in the Canadian Jewish Congress.
In July 2011, Carr was awarded the Order of Manitoba.
In his role as the President of the Business Council of Manitoba, Carr advocated for a temporary increase to the PST by 1% to assist municipalities with their infrastructure deficits.
In 2006 the Business Council of Manitoba president Carr advocated for the province to reduce its health costs, which reached as high as 42 per cent of the provincial government’s expenses, and for government officials to keep an open mind to options that included privatization. "Everything is worth studying, and it's studied everywhere around the world. There are advances that have been made in other places, with cultures similar to our own, that could be applied here," said Carr.
In January 2014, Carr was announced as a candidate for the Liberal nomination in the federal riding of Winnipeg South Centre. He went on to win the nomination meeting and was declared the Liberal candidate for the 2015 federal election. He won the subsequent election on October 19, 2015. Two weeks later on November 4, 2015, Carr was officially sworn into the Cabinet as Minister of Natural Resources.