There are a number of different awards given every year. Some are to a team, but most are to honor individual efforts. Here is a list of the awards, what they are, some historical information on the trophy, and which Colorado Avalanche player, if any, won the award.
Lord Stanley's Cup:
Let's start with the Big One. The Stanley Cup originally started as an award for the top-ranking amateur hockey club in Canada and was donated by Lord Stanley of Preston - the Governor General of Canada. He purchased a "decorative punch bowl" from England, at the time it was worth approximately fifty dollars. Stanley never attended a Stanley Cup game, nor did he ever award the Cup, he passed away before that could happen.
The first winner of the Stanley Cup was Montreal HC in 1893. Originally it was awarded to an amateur hockey in Canada, however, that changed after a trip by the Dawson City Nuggets in 1905. Dawson City challenged the Ottawa Silver Sevens and the Ottawa team was able to beat Dawson City so easily, that it was seen as a farce and something to be avoided in the future. The rules for being allowed to challenge for the Cup were changed. Then, in 1915 the two major professional hockey leagues (the NHA and the PCHA) decided that their respective champs would play each other for the trophy. The NHL was organized in November of 1917, and the Stanley Cup was won by them for the first time in 1918. But the Stanley Cup didn't remain in the NHL permanently until 1926, and it wasn't until 1947 that the league gained full control of the Cup.
There have been two years that the Stanley Cup has not been awarded: in 1919 because of the Spanish Flu and in 2005 due to the lockout.
There are actually three Stanley Cups: the original, the authenticated and the replica. The original Cup was awarded until 1970 and is in the "vault" room in the Hockey Hall of Fame, the replica is also found in the HHoF. The authencated one is the one that is presented to the winning team each year. A difference between the Stanley Cup and the awards given to the other Big Four sports leagues in North America, there isn't a new trophy made every year. Every summer the Cup makes it's way through the players and upper-enchalon of personnel as they each get a day to spend with the Cup as they so choose.
Interesting date: June 17th, 2010. Brent Sopel brought the Cup along with him to the Chicago Gay Pride Parade, in memory of Brendan Burke. This did not count as his official day with the Cup.
Avalanche links: The Colorado Avalanche have won the Stanley Cup twice: in 1996 and in 2001.
Presidents' Trophy: This award is based on the regular season. The team with the most points each season is awarded the trophy. It was first awarded in the 1986 season.
Avalanche links: The Colorado Avalanche have won the Presidents' Trophy twice: in 1997 and in 2001 (when they also won the Cup),
Prince of Wales Trophy: Is given every year, during the playoffs, to the Eastern conference playoff champions. The winner then goes on to the Stanley Cup final against the Western conference champions.
In the past, the trophy was also handed out for: NHL regular season champions, the American Division regular season champions, the East Division season champions, and the Wales Conference playoff champions. The first time the award was handed out was in the 1925 season to the Montreal Canadiens. They engraved their name onto the trophy twice that year. Once for the 1925 season, and once for the 1924 season.
Avalanche links: There are none.
Clarence S. Campbell Bowl: This is the Western Conference's version of the Prince of Wales Trophy. The trophy was made in 1878, even though it wasn't awarded in the NHL until 1968.
Avalanche Links: The Avalanche have won it twice: in 1996 and in 2001.
Hart Memorial Trophy: Is awarded to the player who is the most valuable to his team during the regular season.The winner is determined by the Professional Hockey Writers' Association. The Hart Trophy was first awarded in 1924. The original award was "retired" to the Vault of the Hockey Hall of Fame in 1960 and the new Hart Memorial Trophy took its place.
The voting takes place at the end of the regular season and the writers' rank their chosen Top 5. The Top Three finalists are announced and the winner is given the trophy at the NHL Awards.