A first-round pick by the Islanders (No. 30) in the 2010 NHL Draft, Nelson played two seasons at the University of North Dakota. In 2011-12, his sophomore season, he was third in NCAA Division I with 28 goals and tied for 12th with 47 points. He also helped the United States finish third at the 2011 IIHF World Junior Championship.

Nelson turned pro after his second season at North Dakota. In his first full season with Bridgeport of the American Hockey League (2012-13), he scored 25 goals and 52 points in 66 games, then made his NHL debut in the 2013 Stanley Cup Playoffs in Game 6 of the first round against the Pittsburgh Penguins.

He earned a spot on the Islanders roster for 2013-14 and got his first NHL point with an assist against the Phoenix Coyotes on Oct. 8, 2013. He scored his first NHL goal against the Vancouver Canucks on Oct. 22, 2013 and finished his rookie season with 14 goals and 26 points in 72 games.

Nelson further established himself with New York by scoring 20 goals and 42 points in 2014-15, his first of three straight seasons of at least 20 goals. He scored 25 goals in 2018-19 and four more in the playoffs, helping the Islanders reach the second round.

The Islanders signed Nelson to a six-year contract on May 23, 2019.

Nelson is the nephew of Dave Christian, a forward who won the gold medal with the United States at the 1980 Lake Placid Olympics and later played 1,009 NHL games. His grandfather, Billy Christian, and great uncle, Roger Christian, helped the United States win gold at the 1960 Squaw Valley Olympics.