This may be a bit involved to explain, even if I weren't the one doing it, but try to bear with me; feel free to skip down past the Doomsday Defense for your answer though:

The '70s Cowboys nickname was "Americas Team. " An NFL expansion team in 1960, by 1966 former Giants defensive coordinator Tom Landry had already turned them into a powerhouse as head coach. After the upstart American Football League merged with the NFL the winner of each league played a "Super" Bowl to decide the overall winner, but the first two of these were foregone conclusions: The REAL championship, both times, was played between Landrys Cowboys and former Giants OFFENSIVE coordinator Vince Lombardis Packers. Both games were played on "the frozen tundra" of Lambeau Field in Green Bay, WI (about a stones throw from Canada and bitterly cold in December; Lombardi had special heaters installed before the second game, but when the field cover was removed for the game the field froze solid. ) The Packers won the first after stopping Dallas at their own two yard line on fourth down; they also won the second one, the "Ice Bowl" probably the most famous NFL game ever:

http://en.wikipedia.org/wiki/NFL_Championship_Game,_1967

After that, however, the Cowboys made the '70s theirs to a degree only the Pittsburgh Steelers (four time winners) could contest; of ten Super Bowls, Dallas played in five, winning two but losing two to the Steelers plus another to the Colts. Taciturn Landry stalked the sidelines in his famous fedora assuring us in Christian comics and on Sunday games that nice guys finish first. His clean cut quarterback Roger Staubach (who missed the first four years after Dallas drafted him due to military service, including a tour in Vietnam, after graduating from the Naval Academy) punctuated that with miraculous come from behind wins, one of which produced the Hail Mary pass:

http://en.wikipedia.org/wiki/Hail_Mary_%28famous_play%29

The '71 Doomsday Defense never allowed a touchdown in the playoffs, not even in their Super Bowl win over the Dolphins, who the following year became the only team to go undefeated and win a Super Bowl.

The end of the era was either 1981, when San Francisco made "The Catch" to beat Dallas in the NFC Championship (and go on to win the first of their four Super Bowls that decade) or 1989, when Jerry Jones bought a moribund team from Tex Schramm and casually fired Tom Landry in favor of his college roommate Jimmy Johnson, who'd recently coached Miami to the collegiate championship. At that point the glory days were a distant memory for Americas Team and their fans; only Too Tall Jones and Tom Rafferty remained from the Super Bowl teams (they retired the following year) San Francisco owned the NFL and the New York Giants owned Dallas' division. They hadn't had a winning season since 1985, or a playoff win since 1982. In 1988 they were 3-13 and finished last in their division. A popular joke around here at the time went:

"Would the lady who left her eleven kids at Texas Stadium please come get them. They're beating the Cowboys 10-0 at half time.... "

1989 was even worse: 1-15, with their only victory on a bunch of gimmicky trick plays (including a desperate onside kick to start one half) against the division rival Washington Redskins. But with the first pick in that years draft, they got quarterback Troy Aikman, while Jerry Jones traded standout running back and kick returner Herschel Walker to the Minnesota Vikings for a number of starting players AND multiple draft picks. It's an infamous pivotal deal in which Dallas gave up ONE player and several draft picks (one of which would land Jake Reed, who'll probably be in the Hall as a Viking) in exchange for five players and eight draft picks, including Minnesotas first AND second round picks for 1990 AND 1991. The result would be starting receivers Alvin Harper and Anthony Wright to complement Hall of Famer Michael Irvin, starting linebacker Dixon Edwards, 5 time Pro Bowler and likely Hall of Famer Darren Woodson at safety, plus eight time Pro Bowler, Super Bowl MVP and Hall of Fame running back Emmitt Smith, who holds the NFL record for most yards running in a career.

http://en.wikipedia.org/wiki/Herschel_Walker_trade

The results were electric; the Cowboys were 1-15 in 1989, but in 1992 they were 13-3 Super Bowl Champs over the Buffalo Bills, which is when Jimmy Johnson famously opened his post game press conference with the question shocked fans around the League had been asking all year:

"How 'bout them Cowboys?!!!"

It was no fluke; Emmitt sat out the first two games of the next season demanding a contract that once again made him the highest paid running back (the Bills Thurman Thomas had just signed a deal eclipsing Smiths) and Dallas lost their first two games, but after Troy Aikman GAVE BACK part of his contract (!!!) Emmitt was signed and Dallas repeated as Champs. The following year Jimmy Johnson left and they lost the NFC Championship Game at hated San Fransisco (due to some VERY one sided playcalling, I might add; I still say Tagliabue wanted SF to win the Super Bowl and DIDN'T want Dallas to win three straight, which no one has ever done) but in 1995 they came back again; an NFC Championship victory over the old nemesis Green Bay (this time in Dallas rather than on the frozen tundra) earned them a third Super Bowl trip in four years, and a victory over the evil Steelers earned them their fifth Super Bowl win (SF had become first to five the previous year. )

After 30 years Dallas had played 16 Conference Championships, losing 8 (all but two to Super Bowl Champs) and winning 8 Super Bowl trips, where they won it all 5 times. One sixth of all Super Bowl wins; over a quarter of all NFC Super Bowl teams. According to Wikipedia, "Before the 2008 season an ESPN's Page 2 statistical comparison of all teams since the AFL-NFL merger had the Cowboys narrowly beating out the Pittsburgh Steelers for the top of its Ultimate Power Ranking.[6] " Whether that's true after Pittsburghs record sixth Super Bowl win last year I can't say.