The Cup wins are of course the most memorable. Some of my other favorite memories off the top of my head:
1999-2001 - My favorite Devils teams because they were so complete. The Arnott OT winner is probably my favorite memory of all time, but man 2001 hurts. I really thought that was the best team we had ever iced.
2001-2002 - The return of Domi - Jimmy McKenzie making Tie turtle off the draw, Patty's natural hat trick including tying it with less than a minute left and winning it in OT. One of my favorite games that I can recall.
2003 - MARSHALL! IN OVERTIME! - I was at this game, only an 8th grader and was fading fast. But I rocketed to my feet after that goal, such an awesome moment. Cool story bro moment, I wandered over to the tunnel where the players come out during the overtime intermissions and got high fived by one player, Grant Marshall. Like to think I gave him the magic hands the rest of the way.
2005-2006 - The march towards the division, with that comeback win over Montreal and then sweeping the Rangers. Really fun to watch.
2012 - That whole playoff run, really that whole season, was a blast. I loved that team and was so disappointed with how quickly they disbanded. One of my favorite underrated moments of that playoff was Poni's OT winner in Philly. It had never felt louder in that arena, at that time at least.
Just the pleasure of getting to watch Marty Brodeur for virtually his entire career. My favorite player of all time. Stevens, Nieds and Patty all fall into this category as well.