The game was released as a launch title for the Nintendo 64 and together with other titles like Ridge Racer, Tomb Raider and Wipeout on the PlayStation it truly defined the 3D revolution. Nothing like this had ever been seen before. Players got to explore the familiar world of the Mushroom Kingdom in glorious 3D, and the order in which the levels could be tackled was largely up to the player. Although each star and world was numbered, if you knew what you were doing you could collect them out of sequence.
The tasks involved in obtaining stars were many and varied, from racing Koopas, defeating giant bombs, collecting 8 red coins, reuniting baby penguins with ther mum's and many more. In total, the original game had a huge 120 stars to collect in total, although only 70 were required to face Bowser for the final time and complete the game. The DS version adds 30 new stars, and to get all 150 will take you many hours indeed. Whereas getting every star didn't really reward you with very much on the N64, in the DS version you actually get something for your troubles (a new mini game).
The N64 console, and this game in particular, also introduced players to the analogue stick (which Sony would later adopt for their Dual Shock controller). This took some getting used to, but after a while the precision of the controls became second nature. Of course, the DS has no analogue stick, and to get around this fact the touch screen and stylus are used instead. Many people have complained that this is a poor replacement, but I had no problem with it, and it many respects it is actually better (especially for swimming or flying).
Super Mario 64 has still (arguably) not been bettered to this day. Not even Nintendo themselves could come close to bettering it with Super Mario Sunshine. There are many good 3D platformers out there that are worth playing, but none of them have the same level of imagination and sheer wealth of things to do in the same way that SM64 has.
Opinion is divided on which is the better version of the game. For the purists, the original will always be the best, while those who missed the N64 will no doubt prefer the DS version. With the addition of 36 mini games, 30 stars and a multiplayer mode, it's no simple cash-in port (that many people feared it would be). If for some inexplicable reason you still haven't experienced the masterpiece of SM64, either pick up an old N64 or a new NDS, and prepare to be dazzled. As the great Dr. Jones said, it belongs in a museum!
Bowser on the N64.
Mario on the NDS.