Krump is an urban dance style that developed on the streets of South Central Los Angeles around 2001-2002 and has since spread around the world. It began as a non-violent way of releasing aggression and anger in areas where street violence was widespread.
It represents a major part of the hip hop culture and it is constantly evolving. You can get a better idea watching the documentary Rize directed by David LaChapelle, which depicts the street dance styles of krumping as well as clowning. The documentary has given krumping plenty of exposure and since then the dance style has risen in popularity in the hip hop dance community.
Clown dancing is an early form of krumping that started in the early nineties with a Los Angeles performer named Tommy the Clown. He is well-known for his trademark dance style but also for his colourful clown wig and make-up. Many dancers who are into clowning include white facepaint as part of their performance. Clowning moves tend to be less aggressive than those used in krumping.
There are many different styles of krumping ranging from the more purist (like "goofy", "rugged" and "monster" styles) to the more edgy and aggressive (as seen in "beasty", "cocky" and "flashy" styles), you just need to choose yours or mix them up and make your own unique style.