Mongo is a style of pushing a skateboard, also known as "spearfoot(ed)". When not pushing mongo, the front foot stays on the board a little bit behind the front trucks (unless you're riding fakie). When pushing mongo, however, the back foot is the one that stays on the board, so the board sticks out in front of you like a spear.
Pushing mongo is neither better nor worse than the alternative. Many very good skaters skate mongo, including Bill Danforth and Bam Margera. Even so, many people consider it an "incorrect" way to push a skateboard.
No one is sure who was the first person to push mongo, since many beginning skaters do it instinctively.