In chronological order:
Wins:
*1988 regular season finale vs Chicago
*1995 Eastern Conference Finals Game 5
*1995 Stanley Cup Finals Game 4 clincher
*2000 Eastern Conference Finals Game 7 (ironically, this game may have been even bigger as a negative in Flyers history than a positive in Devils history)
*2000 Stanley Cup Finals Game 6 clincher
*2003 Stanley Cup Finals Game 7 clincher
*2012 Eastern Conference Finals Game 6 clincher
Losses:
*1994 Eastern Conference Finals Game 7
*2001 Stanley Cup Finals Game 7
All of the games except the 1994 game have already been mentioned, so let me add a few thoughts about that game and series. IMO, though the Devils lost to the hated Rangers (who broke their 54 year Stanley Cup championship drought a couple weeks later), it was the greatest playoff series in NHL history, one so good a book has been written about it (by a Devils fan to boot!). Even more importantly, even though the Devils lost, in many ways that series and season established the Devils as one of the best franchises in the league, and established the Devils as a legitimate franchise in New Jersey. People didn't expect the Devils to win that series against the Rangers, who had A LOT more to lose than the Devils did, and when the Devils played their Manhattan rivals, who had the best record in the NHL that year, so evenly for 7 games, it validated the step forward the Devils took during the regular season that year (they finished with the second-best record in the NHL behind the Rangers). What happened in 1995, when the Devils had an indifferent regular season but got themselves geared up for the playoffs and then had their great playoff run, really had its context set by what happened in 1994. I'm not sure 1995 happens without what happened in 1994. Game 7 of the 1994 ECF was a heartbreaker at the time, but especially in retrospect it was definitely more of a beginning than an ending, the beginning of the Devils' run as one of the elite NHL franchises over the next decade.