The Palm Beach Post explains how all the NHL teams got their nicknames. Here's how the Boston team came to be known as the Bruins:
The team was founded in 1924. The owner, Charles Weston Adams, also owned a local retail chain whose colors were brown with yellow trim. A naming contest stipulated the nickname would be "an untamed animal whose name was synonymous with size, strength, agility, ferocity, and cunning; and in the color brown." (A bruin is a type of bear.) The uniform front shows a wheel, which is believed to refer to the hub configuration of Boston's streets.
And here's how Montreal became knowns as the "Habs":
When the National Hockey Association was founded in 1909, Montreal already had a team comprised of English-speaking players called the Montreal Wanderers. A group then founded a team of French-speaking players and called it "Les Canadiens," French for Canadian. The NHA became NHL in 1917. "CH" on front of Canadiens uniform is French for "Club de Hockey Canadien." In 1920s, New York Rangers owner Tex Rickard had picked up on a rumor that the "H" was for "habitants," a French slang term for a Quebec farmer. The nickname stuck and was later shortened to "Habs."