We don't call them sides, we call them teams, but here's some of the best:
Baseball: The Yankees. They've won exactly 1/4 of all World Series (US/Canadian championship, but mainly US, since there have only been 2 Canadian teams and currenlty there's only 1). The Yankees have won more championships than any other team in the history of Professional Sports, with 26. The Yankees clinched their division yesterday and currently have the best record in baseball, preparing for the play-offs starting next week. That leaves everyone else in the dust, the number 2 winning team having only 10 championships. Also, nobody with more than 3 appearances in the world series can beat the Yankees winning percentage of 2/3 (26/39). Precisely half (52 out of 104) of all World Series have included at least 1 team from New York. 14 times, both teams were from New York. The Yankees have, in different eras, won 4 in a row, 5 in a row, and 4 out of 5 MLB championships and are division champions again after last year snapped a record 14 trips to the post-season in a row.
Hockey: The Montreal Canadians 23 NHL championships made them the top team in professional sports history until the Yankees passed them in the 1990's. That would probably make them the best team in Hockey. I'm not sure who the best American team is, but the Americans play in the same league as the Canadians, so you can count them together. It's really a bigger sport in Canada than in the US anyway.
Soccer: The US National Women's team is the best. Major League Soccer hasn't been around very long in the US, but thanks to the Olympics, there is a long tradition of National teams, often including many college players who are not professional. The Women's team has been competitive at an international level for a long time. The Mens National Team has only recently become competitive on an international level. No MLS team can really claim any sort of dominance though, the league is just too young for that.
Basketball: There are three major dynasties in Basketball, the Lakers, the Bulls, and the Celtics. I'm not enough of a basketball fan to really say who is the best among the three, but I couldn't imagine a team better than Michael Jordan's Chicago Bulls of the 1990's with Scotty Pipen and Dennis Rodman among others. Of course Jordan shined above them all making the impossible look easy.
Football: The Steelers, Cowboys, and 49ers are among the best in the NFL. The Cowboys have played in 8 Superbowls, winning 5 of them. The Steelers lead the NFL with 6 Superbowl wins, having lost only once, giving them 1 less appearance than the Steelers, but 1 more victory. The 49ers, however, have an impressive 5-0 record in the Superbowl, having never lost. The Steelers are the reigning champions. Also of note, the Patriots have made it to the Super Bowl 6 times and boasted a perfect record in 2007 until losing in the Super Bowl to the New York Giants. The Giants 3-1 record isn't as impressive as some of their competition, but they did score two of the most impressive victories in Super Bowl history.