Here's a timeline of some of the notable milestones:
1948: Larry Kwong appears for the New York Rangers; he was the son of Chinese immigrants to Canada.
1958: Willie O'Ree, playing for the Boston Bruins, is the first black player to play in the NHL. He has been called the Jackie Robinson of the NHL.
1989: Robin Bawa, the first Indian player, joins the Washington Capitals.
1989: Bill Guerin becomes the first Hispanic player in the NHL. One of the top scorers ever in the league, he will likely be elected to the Hall of Fame as soon as he is eligible. He won two Stanley Cups in his 18-year career, during which he played for eight different teams.
1990: Jim Paek is the first Korean player, playing for the Pittsburgh Penguins and eventually winning two Stanley Cups with them.
1994: Paul Kariya is the first player of Japanese descent as he joins the Anaheim Mighty Ducks, and later becomes the first player of Asian descent to score 50 goals in a season.
2003: Jordin Tootoo is the first Inuit player in the league. He is still playing for the Nashville Predators as of this writing.
Willie O'Rea, Hockey Pioneer [pdf]
List of ice hockey players of Asian descent
List of ice hockey players of Middle Eastern descent (not used as categorization is difficult)