I think the DS and the DS LIte cartridges are different shapes and sizes, but the DS Lite has two slots allowing it to accept either type. I've only ever played the "Brain Training" type games and so only ever tried the later (smaller?) DS Lite cartridges.
As to games there is Big Brain Academy, Sight Training, Brain Training II and Picross all of which I've tried and enjoyed. Sight Training is my current favourite out of these. Of the older type games, when my kids were younger Super Mario was all the rage and I never did adjust to the Nintendo console pad. I was a mouse/ keyboard PC gamer (if at all) and was regularly beaten at anything I tried that started with the word Nintendo. Maybe I should have a secret try at Super Mario Kart after all these years and see what I've been missing?