I can give you some broad strokes, because really what defines the Patriots is that they can morph into whatever they need to based on how they want to attack/defend the opponent.
On defense the most basic way I pin down their base is the Double Nose. Most of the 2000’s they were primarily a 3-4, with the three lineman two-gapping, allowing their diverse group of linebackers to make plays. This was derivative of the Fairbanks-Bullough 3-4 system.
But as the game evolved they went more to a two linemen look with edge players like Ninkovich and Jones being hybrid players. This was an effort to get more athletes on the field and be better against the pass in their base defense. So now you’ve almost got more of a 2-5 front, though who’s one-gapping and who’s two-gapping will change from snap to snap.
But as we’ve often said, nickel is really the new base, and the Pats have spent a great majority of the last few seasons in the sub-package with at least three defensive backs on the field. Their sub-packages are usually fairly straight forward, with a dime linebacker role (Money) usually played by a big safety.
The overall defensive philosophy of Bill Belichick is a conservative one. He’s never had an attacking defense featuring constant and complex blitzes like Dick Lebeau or Rex Ryan. Instead, he prefers his defense to play sound assignment football and win the one-on-one matchups. This is why you often hear “bend-don’t-break” to describe the Pats defense. They’ll force quarterbacks to consistently execute and more often than not that leads to turnovers and mistakes.
Offensively the Pats playbook has evolved from the Erhardt-Perkins system. If you google that with “PatsPropaganda” you’ll find plenty more info on the history of that system. Of course ,now it has become essentially the Tom Brady offense, but the core tenet of it is the short, controlled passing game. Simply put, it’s “throw to score, run to win”, though the offense is no longer the smash mouth one that gave the system it’s reputation in the 80s.
Most important to the system are the route combinations that can often change pre and, even post, snap. As opposed to systems where it’s a simple route call for the wide receiver and he runs that route regardless of the defense, the Pats’ system requires the receivers and backs to read and understand the defense. It’s hard to coach and execute, but once in place, it’s the hardest to stop because you have six thinking and adjusting chess pieces instead of just one in the quarterback.
The Patriots will run a core group of plays, but disguise them with different personnel and formations. When they had two tight ends that could do everything, it made the offense even harder to stop because they could seamlessly shift between the power game and a spread offense.