I do log on and read this site but I made an account today especially to comment on this thread.
I believe the greatest all-time rivalry in hockey is Russia-Canada. While I wasn't there to witness it in the beginning, I've definitely heard about it plenty and watched video clips and read about it. While it was politically-charged and bitter to the bone in the beginning, I do believe it has changed over time and become more of a healthy respect for another intense hockey nation.
I actually love meeting Russian hockey fans online because I really value the great history we have together in hockey and I visit The Big Red Machine page on FB and exchange well wishes during the world juniors this year and last, as well as during the Olympics. I remember last year, I asked the members there (who appear to be primarily in Russia) whether they would prefer to meet Canada or the U.S.A. in the gold medal game for the Olympics (because I had seen some Americans posting other places that there was no question that the Russians would want to play them because of the 1980 Olympics.) Almost all the responses I got back said they most wanted to play Canada for the gold because of the history and the classic rivalry/intense games. Even a couple days ago, I posted on the page again saying that I hoped we'd be playing them for the gold and the page owner said quite enthusiastically that he wished to play Canada for the gold every time. And btw, Tretiak himself went on record at the Olympics saying he wanted a Russia/Canada gold medal game too, it was on video.
Lastly, I know that if you are talking about the political arena or sports at large, then the answer would be different, but I do believe the forum is about hockey and I'm answering from a hockey only perspective.
Second rivalry for me would be Canada/U.S.A. (and I know this will mean less to non-North Americans but I can only speak from my own perspective.) However, I'd have to say that I echo the other poster who said that he'd rather lose to Russia any day than to the States. I personally feel there is a lot more animosity between American and Canadian hockey fans (not all mind you, but a significant portion can get quite nasty with the comments when it comes to international competition and insist on bringing in totally non-hockey related insults or hockey-related ones that make absolutely no logical sense) and I also get tired of "who cares about hockey anyways! that's all you guys are good at, we dominate in every other sport, why don't you try to beat us in a REAL sport like football!" etc. etc.type reactions when we beat them.) I just think it's a pretty unhealthy relationship I'd say... maybe ok if you are just discussing NHL teams but definitely not the healthiest when discussing international competition. And again, I am not saying all Americans fall into that category and there are jerks on both sides I know, but just saying generally I don't like dealing with all that animosity.
I know Sweden & Finland are a big rivalry too but I'm not European so I would leave the European rivalries to the experts.
Sorry for the longish post.