It's Mario, almost exactly as you remember it.
The game's plot, as mentioned before, is rather thin (much as it was in the original Mario games). Bowser Jr. kidnaps Princess Peach literally from behind Mario's back. From there, it's up to Mario to chase him down and rescue the Princess. The game's levels are divided up into eight different worlds based on various themes such as a tropical area with lots of water stages and a sky world where Mario will have to be careful not to fall off of the clouds. Two of the stages can actually only be reached through a branching path, so going through every single level actually isn't required to beat the game. However, this game's incredibly tight level design makes playing through all the different stages so much fun that people who really like it will no doubt want to play through every stage at least once.
The big deal about this game is, naturally, how it's a return to the side-scrolling gameplay that made the Mario series such a big success in the first place. Nearly every stage involves taking Mario from left to right, although there are a few exceptions during the castle stages. This actually makes it feel closer to the very first Super Mario Bros than any of the (relatively) newer games. A few new moves have been added from Mario's three-dimensional adventures to make things interesting, such as his wall jump and his triple-jump. Power-ups are mostly limited to things that were seen in the original game, like Fire Flowers and Super Stars. A few new powerups have been added, like a blue Koopa Shell that Mario can wear to slide along the ground like a Koopa Troopa, but it's not nearly as common as any of the others. There's also a Mega Mushroom, which makes Mario incredibly large and able to crush almost anything in his path, and a Mini Mushroom, which makes him incredibly small and able to fit into places a larger Mario wouldn't.
In addition to the main game, there are several mini-games available from the main menu, all of which use the DS touch screen and or microphone in some way. Many of them originally arrived on the DS rerelease of Super Mario 64, so they should be familiar to anyone who played that game. Most of them are fun as a diversion, but they're not the selling point of this title and they certainly aren't the best part. There's also a wireless multiplayer mode for players who want to play a sort of side-scrolling scavenger hunt between Mario and Luigi against another person.
The game's graphics appear to be an attempt to modernize the series' classic look. Mario and many of the enemies look like 3-D polygons instead of the sprites of yore, and most of the characters look like the renderings that you see on the box or in the game's artwork. It's a cool look, although most of the characters don't look completely perfect. There are a few neat little details to the game's visuals though, like how some of the game's enemies move in time with the soundtrack as it plays.
Sound is another highlight. While there are more than a few references to old Mario tunes, there are several new songs that are very well-written and incredibly catchy in their own right. Plenty of voice samples have been added to this game as well, something that has happened to nearly every Mario game since the Game Boy Advance was released. It doesn't seem quite as out-of-place here, since this is a new game and not a remake of a game that had none of these.
No experience with the older 2-D Mario games is necessary to enjoy New Super Mario Bros, as it does a spectacular job of standing on it's own. However, anyone who has played those games will instantly feel comfortable with this one, as it does a spectacular job of picking up where all of them left off. And with the high quality of all the different stages that have been put into this game, it's the sort of title that anyone who enjoys it will want to play though again and again. This game is a must for anyone with a Nintendo DS.