The game begins with Mario and Princess Peach taking a walk outside the castle. Suddenly, while Peach's Castle is in view from them, a cloud goes over the castle, and lightning starts to strike it {possibly Lakithunder}. All the Toads start to scream and jump out of the castle, and Mario goes over to help, expecting that Peach would be okay. While Mario runs over to the castle, Bowser Jr. sneaks up behind her, and kidnaps Peach. Mario looks back, realizes this, and starts to run toward him. Bowser Jr. will then throw a Koopa Shell at Mario making him Small Mario and hide inside the nearest Tower with Peach.
Mario now goes throughout the grassy land toward the Tower. Once he gets there, Bowser Jr. will decide to fight Mario. Once Mario wins, Bowser Jr. escapes, takes Peach, and runs off, Mario will now have to go through many obstacles to get her back. While trying to find her, Mario will encounter a castle, and finds out that inside waiting isBowser. Mario will get past him, and jump on the switch to break the bridge that Bowser is on {like in the original Super Mario Bros.}, whom will fall in the lava and turn into a skeleton, turning him into Dry Bowser.
Mario will have to now go through the Mushroom Kingdom tracking down Bowser Jr., going through deserts, oceans, jungles, ice lands, mountains, and even the sky. Once he gets to the last land, the Lava world, he will encounter Dry Bowser. You will have to get past all the bones that he throws, and the monster himself, to hit the switch to once again make him fall through a deep bottom pit. However, he can still spit fire. He is somewhat slower than Bowser.
Now, Mario goes out to find Bowser Jr again, and comes up to a huge castle, which he enters in hope of finding Peach. Once he gets to the end, he finds Bowser Jr., whom will throw the Bones Bowser into a pot with potion that revives Bowser. Now, the two will fight you, where you'll have to get past both of them to hit the switch, which will drop the bridge. Now Peach will come down off her platform, and kiss Mario on the cheek, which ends the story.
source:
http://nintendo.wikia.com/wiki/New_Super_Mario_Bros.
source: google image