First memory is a friend of a friend getting an import version. He brought it to one of our gatherings of some sort, and I was utterly amazed at how small (and heavy) it was. When you held it in your hands, you felt like you were holding this console of the future.
Short time later I imported my own with VF3 and Sega Rally 2. Having games like that at home was just mind-blowing, especially since we were coming off of the first real generation of home consoles doing 3D. What the Dreamcast was doing was just light years beyond the PS1, Saturn, or N64.
I remember getting my Interact (I think it was) VGA box, and hooking the console up to my nice Sony Trinitron monitor. Long before the days of true HDTV, I got a taste of just how beautiful games could look.
Phantasy Star Online. Ah, the memories. I'm not sure even WoW has caused me to lose as many hours as that game did. It was my first real taste of an online gaming community within a game, and the hours I spent running Ayu Landale through missions with people can never be counted. Deciding that I most favored the Partisans, finally getting a hold of the elusive Angel Wings mag, being annoyed by having to run those fuckings Caves yet again, it's utterly amazing to think back and realize just how much I got from that game from how little content there really was. (Playing WoW + BC expansion, I'm starting to feel like the world is a bit too small, but damn it's like eternity compared to the amount of areas in PSO.) I'm not sure if I'll ever be addicted to a game again like I was PSO... it was the right game at the right time.
The BBA. Man, how Sega of America royally screwed up the release of that thing in the US. You had to order it from them directly, and the botched the launch like nobody's business. I finally ended up getting mine (and still have it), and it make my PSO experience much better.
The VMU. The green Godzilla VMU was the very first piece of anything Dreamcast released, and I actually imported one. There wasn't much to do with it - especially since I knew nobody else who had one - but it was just this amazing little device that, again, felt like the future. I then remember when people started learning how to make your own VMU software: I made a little "application" called NekoKana where it would randomly show you a Japanese katakana character, and pushing any direction on the d-pad would then tell you what that character was. You could push the d-pad again to once again look at the character to help memorize it, or push like the A button to randomly bring up a new character. Even though I wasn't a programmer, here was something I had created running on a piece of gaming hardware - amazing.
The games, of course. Nevemind the big names, you had an amazing selection of some of the best all-time Capcom and SNK fighters (KOF '98, Last Blade 2, Mark of the Wolves, SFIII, Project Justice, MvC 2, Capcom v SNK, etc.), the first truly 3D Resident Evil, Mars Matrix (the shooter that actually made me like a shooter), Coaster Works, Metropolis Street Racer, Get Colonies, Chu Chu Rocket (aka the "make your friends hate you" game), and much more.
I've tried getting rid of my Dreamcast, but I just can't do it. Even though I've gotten rid of a few pieces - Ikaruga, one of my sets of maracas, my copy of Samba de Amigo - I've still got an insane amount of games, accessories, and whatnot for the console. Honestly, I don't know that I can ever get rid of it; I was a Sega person until the end of the Dreamcast, and being their final system (until next year, muuhahaha!), I just can't part with it. It was really the end of an era, and a damn fine effort to go out on.