In the NHL one of the oldest and probably the most bitter rivalry is the Leafs and Canadiens. It brings a history of English Canada vs French Canada, which is one of the reasons why it is intense, although the English vs French thing probably isn't as relevant today. But every time the Leafs and Canadiens meet, there is always a great energy between the teams and the fans.
So as for the biggest, I would go with Leafs-Canadiens. But there are many more rivalries in the NHL that are pretty big and intense as well, ie. Canadiens-Bruins especially and one that many could argue for being the biggest; Oilers-Flames, Red Wings-Avalanche etc. Also the battles of Provinces and States, Leafs-Senators, Oilers-Flames as mentioned already, Flyers-Penguins, Rangers-Islanders etc, some of these not as big as others.
For international men's hockey, its Canada-Russia. Look up the Summit Series. Its always something special to watch the best of these two countries go at it. At the Vancouver 2010 Olympics, there was huge hype for the two to make it the finals and face off against each other. The finals ended up being Canada and the US though, which could show that maybe another big rivalry has emerged, as the two countries have met twice in this last decade (2002, and 2010) in the Olympic Gold Medal game. To add a little more fuel to this rivalry, the US beat Canada at the World Juniors on home soil, ending Canada's dominance at the World Juniors and attempt for 6 straight World Jr Championships.
For international women's hockey, it's Canada-USA. Since Women's Ice Hockey was introduced at the Olympics, the two have met in the finals every singe time. The US won the first Gold Medal, with Canada winning the next 3. For the 2010 Olympics, I believe the US had won every tournament, and if I'm not mistaken, every game against Canada leading up to the Olympics. The gold medal game in 2002 was pretty intense as well. The American had the Canadian flag on the dressing room floor, and the officiating was fairly one sided for the Americans, however Canada still managed to pull off the win in an amazing game.
In 2002 and 2010, the finals were the same for both men and women at the Olympics, Canada vs USA, showing that this rivalry is definitely alive and strong. Some argue that for the men, the rivalry today could be just as big as the rivalry with Russia.
The Montreal Canadians. Those 2 teams were always fighting for the Stanley Cup.
Probably the Boston Red Sox are the Yankees biggest rival.
blackhawks,blue jackets,predators,red wings.
Comcast is their biggest rival.
The UNITED KINGDOM was Napoleon's biggest national rival.
Ben Johnsons biggest rival was Carl Lewis
Hurricane Grill & Wings just one the festival in Buffalo for 2007. They have no rival.
Red .... wings. Wings. Do you get the significance now?