Born in Haliburton, Ontario, Nicholls grew up in the nearby hamlet of West Guilford. He scored 79 points as an OHA rookie with the Kingston Canadians in 1979-80. That summer the Los Angeles Kings at the NHL Entry Draft chose him 73rd overall. After attending his first pro camp, Nicholls tore up the OHA with 152 points in 1980-81.

The transition to the AHL was an easy one for Nicholls as he scored 41 goals in the New Haven Nighthawks' first 55 games before he was recalled. His impact was immediate as he scored 32 points in 22 games then scored four goals in the post-season when L.A. upset the Edmonton Oilers and reached the second round. In 1982-83, he experienced a few bumps along the way to a 28-goal season. The next year he became a bona fide star with 95 points and hit the 100-point mark for the first time in 1984-85. Nicholls was also an offensive leader for Canada when they won the silver medal at the 1985 World Championships.

Nicholls continued to be a consistent scorer for the Kings and one of the NHL's most dangerous threats while shorthanded. In the aftermath of the acquisition of Wayne Gretzky in 1988-89, Nicholls set career-highs with 70 goals and 150 points while finding the net 21 times on the power play. By the next season the Kings sought to add a bit more chippiness to their team to take them to the next level. As a result, Nicholls was traded to the New York Rangers for gritty and talented forwards Tomas Sandstrom and Tony Granato. During the 1990 playoffs Nicholls scored 12 goals in ten games while helping his new club reach the second round.

Early in the 1991-92 season, Nicholls was the key player acquired by the Edmonton Oilers for Mark Messier. Injuries reduced his participation to 49 games during a disappointing regular season but the veteran responded with 19 points in 16 playoff games when the Oilers reached the semi-finals.

By the start of the next season, Edmonton was rebuilding and traded Nicholls to the New Jersey Devils. He adapted his game to the tight-checking system of Jacques Lemaire and helped the club come within one game of reaching the finals, in 1994. A few weeks later he signed with the Chicago Blackhawks and averaged over a point per game for the first time since 1990-91. He was at his best once again in the post-season when the Hawks reached the semi-finals. Nicholls played one more year in the Windy City before signing with the up-and-coming San Jose Sharks in August 1996. The veteran forward added experience and playmaking ability to the young team before retiring at the end of the 1998-99 season.