This game is multi-player (up to four players at a time). In other words, it's 1-4 player (pl-1 plays as Mario, pl-2 plays as Luigi, pl-3 plays as a yellow Toad, and pl-4 plays as a blue Toad). Plus, You can ride Yoshis in this game (Green Yoshi for Mario, Blue Yoshi for Luigi, Yellow Yoshi for Yellow Toad, and Pink Yoshi for Blue Toad)!
Here's the story (WII only).
Princess Peach's birthday and as Mario, Luigi and their friends are celebrating, a huge cake slides in. Immediately, Bowser Jr. and the Koopalings pop out and throw the giant cake on top of Peach, trapping her inside. From there, the villains load her onto their ship while Mario, Luigi, Blue Toad and Yellow Toad follow closely. The Toads from the castle soon use a cannon to blast away the Propeller and Penguin Suits towards Mario and the others so they can have access to them.
When Mario and the gang finally get to the final castle, they find none other than Bowser himself. One of them has to jump on a ! Switch behind Bowser to defeat him, sending him falling into the lava below. After, Mario, Luigi, Yellow Toad, and Blue Toad celebrate their victory, they see Peach crying in a cage. However, she turns out to be Kamek in Peach's dress. Kamek then powers Bowser up, making him huge. After Bowser has been powered up, the four can do nothing but flee from the giant boss. Soon they find a dead end with a huge ! Switch. They push it, and Giant Bowser falls through the floor in final defeat as Princess Peach is released from her cage. As they venture out, they see Luigi, Blue Toad and Yellow Toad arrive in their hot air balloons. Luigi lets Mario and Peach into his balloon and is accidentally left behind. The two Toads then offer Luigi a ride in their balloon. As they sail off, Peach asks Mario if she told him about the Secret World.
Larry Koopa is seen limping toward Bowser's Castle. He meets Bowser Jr., who shows Larry the rest of the Koopalings trying to push Bowser, who is back to normal, right side up. They finally succeed, with a huge thud, causing the castle to fall on top of Bowser and his children 
Here's the story (DS only).
Mario (or Luigi depending on who the player is playing as) and Princess Peach are walking together around Peach's Castle, when a mysterious thundercloud appeared, and struck the castle with lightning. Mario/Luigi went to investigate, and while he is there, Bowser Jr. sneaks behind a bush and kidnaps Peach. Mario/Luigi follows, and Bowser Jr. drags her behind him through eight different worlds. The player must save Peach.
Dry Bones-like version of himself (known as Dry Bowser). Later, at World 8, Mario has to face this undead version of Bowser as the penultimate boss in the game. Dry Bowser falls into a pit, but Bowser Jr. somehow finds his father's bones, and he revives him by tossing his remains into a bubbling cauldron, which results in not only Bowser returning from the dead, but growing roughly three times his size.
At the end, Bowser and his son fight Mario together, but Mario defeats them, sending the two into yet another pit. After the ending sequence, Bowser Jr. is shown dragging his unconscious father across the castle floor. When he notices the player watching him, he shouts something and then continues on with his task.