Such contests have been defining of the Bill Belichick era — generally serving to establish the AFC pecking order — and sometimes even featuring teams sporting undefeated records.
The Patriots, Jets and Colts, however, all offer very different challenges. Games against New York tend to be of the gritty variety — defensive contests in cold weather. New York hasn't had the offense to keep up with New England in recent memory, but they always bring defensive schemes that throw a monkey wrench in the Tom Brady offensive machine. It also should be noted that the Pats get the Jets twice a year, so the second contest is always a revenge game.
With the Colts, the matchup is always about Brady versus Peyton Manning. The two best quarterbacks of the generation annually have an opportunity to state their case, and they rarely disappoint. The games tend to be shootouts of the most outlandish varieties — no matter how each team has been playing going into the contest.
The Chargers, of course, are the third spoke of the "Chargers beat Colts, Colts beat Pats, Pats beat Chargers" Rock-Paper-Scissors game that has occasionally unfolded over the years. With Philip Rivers — and formerly Drew Brees — the Chargers always feature what can be argued is the league's most prolific offense, and the games tend to be full of big plays and turnovers.
Of course, these aren't the only three teams that the Pats have rivalries with. The Steelers, Ravens and Dolphins surely are teams that Pats fans relish victories over.