As usual, Mario is the most well-rounded character. He's a great character to use when you first explore a new area.
Luigi can jump much higher than any other character. Choose Luigi when you're tackling an environment with lots of high platforms.
Peach is not very strong or fast, but she possesses the poser to float through the air for a brief period. This ability makes her an excellent choice for areas with dangerous jumps, conveyor belts or ice.
Mushroom-headed Toad is extremely fast and very strong. He can pick up items much more quickly than his fellow adventurers.