It changes over time, but the Packers-Bears is the oldest.
I think for a rivalry to be good, you have to have two teams that really want to beat each other, and there has to be a chance for either team to win, most of the time. In the seventies, it was the Cowboys-Redskins and Oilers-Steelers. These teams hated each other. A few times they were trying so hard to hit each other they forgot to play football. Funny thing, the Falcons and Saints had a heck of a rivalry in the 70s, cause they weren't very good, and they had a chance to win. Those guys had some good games.
In the 90s, it was Cowboys-49ers, and the best rivalry of the new millennium has been the Colts-Patriots, cause the team that won those games usually won the Super Bowl. I think the Cowboys-Steelers is the only real Super Bowl rivalry, cause they've played three times.
For us, I think we're the top rival for all the other teams in the East. The Giants-Eagles also has some bad blood.
Although a lot of people think the Packers have gotten the best of us in the playoffs because of the Ice Bowl, we actually lead them I think 4-2 in the playoffs, including putting them out three straight times in the nineties. We used to have a heck of a playoff rivalry with the LA Rams, and they nailed us a few times in the playoffs. Dittos for the Vikings, back when they were an outdoor team and had that frozen north toughness. Now, they're a dome team. Man, those guys looked tough standing out there breathing that smoke.