Mud trucks are a type of 4x4 trucks. 4x4 trucks are 4-wheel drive trucks that get super traction and have more power than 2-wheel drive trucks, or 4x2 trucks. Because the vehicle supplies power to all 4 tires, they operate better and are easier to maneuver over tricky terrain than trucks that only supply power to 2 tires.
Mud trucks are designed for one purpose--mud racing. Mudding trucks aren't always used for racing, though. They're often used for just exploring wet and rough terrain by those who like to get outdoors and off-road.
Mud racing, more widely known as mud bogging, is sport that's popular in North America for those who like to take their 4x4 trucks off-road and compete, and who don't mind getting muddy in the process. A large mud puddle, sometimes a very long mud pit, is found or created, and usually measured for depth. Then each driver tries to get his truck through the mud as quickly as possible.
It's not uncommon for one or more mud trucks to get stuck in the mud and be unable to get out, so a tow rope is a necessity at a mud race. Some mud trucks even come equipped with winches and ropes to make it easier. The mud truck that gets through the mud in the fastest time wins.
4x4 mud bogging trucks are a little bit different than standard 4x4s. These trucks are usually bigger with bigger tires, and the tires are extra ridged and treaded for better traction in the mud. True mud lifted trucks are harder to drive than regular trucks or standard 4x4 trucks, and can require special practice and training.
Mudding trucks may have special reinforcements in the cab and chassis, and special protections like rollover bars and other features can be added to protect the driver and/or passenger in case the truck rolls over. This isn't out of the realm of possibility in mud bogging, because sometime the combination of steep or hilly terrain and slick mud can cause rollovers and slides.
That's why you shouldn't try mud bogging without an actual 4x4 mud truck. It would be nearly impossible to do this in a 4x2, not to mention how dangerous it would be. Even in a standard 4x4 that might be able to handle the terrain, you won't be as well protected in the event of a roll or a mishap as if you would be in a mud truck.
You can find a variety of 4x4 mud trucks for sale online. Keep in mind that these trucks, since they're considered racing trucks and sporting equipment, are usually painted brightly with lots of customization so they're going to be more expensive than even a custom 4x4 truck. You also may be able to take your 4x4 and customize it into a mud truck by lifting the chassis, getting bigger tires and making safety and performance improvements. Check with the American Mud Racing Association for information about what needs to go into a mud truck.