I only recently received a DS Lite (last Christmas, literally everyone else in my family owns one...) and have been playing a backlog of games originating with when the original launched. In my opinion, there have been four games that stick out to me as what the DS library should represent.
1. Meteos: I adore puzzle games and figured this genre would be DS' strong point. This gem of a title is fresh and frantic in its play style and uses the stylus well (especially the flick blocks up aspect). Lots of features and unlockables to keep you playing. Note: there is a slight "hold L + scribble" abuse that may ruin the game for some, but one should avoid it if you have your strategy in order (except on 100 meteo war, ha!)
2. Kirby Canvas Curse: I believe, far and away, KCC demonstrates the direction platform games should have taken on the DS - its also the game I've enjoyed most. Its almost like a mix of... NiGHTS, Lemmings and a pinball game, I dunno, that really doesn't do it justice. Its control scheme should have been as revolutionary as Mario 64 was to 3d platformers, but lo and behold, I haven't seen anything nearly as creative or enjoyable since.
3. Elite Beat Agents: The touch screen immediately made me think of rhythm action games. The unique touch sensitive features (not just timing your presses, but rolling balls etc.) make it stand out beyond the normal "time your button press with this key" fare. Furthermore, the presentation, comic value and quirky nature push it over the top.
4. Brain Age: These brain training exercises are addictive and entertaining. I can't believe an entire genre blew up over this idea so quickly. My father still plays his DS everyday with Brain Age. My entire family is competive with the fastest times and constantly compares our scores. The funniest aspect is comparing pictures, where we'll leave messages to each other, poking fun at our last competition. Fun for everyone!
Hope you enjoy!
-Dr. Ugly-