Pushing mongo is the practice (or malpractice) of pushing with one's front foot, that means pushing with your left foot if you are regular or your right foot if you are goofy.
People who push mongo are usually beginners or otherwise inexperienced with skateboarding and will stop doing it after learning some of the basics of skating.
Pushing mongo is disadvantageous for a few reasons, the distribution of weight towards the tail of the board means it will be more difficult to maintain balance, additionally it makes certain tricks such as the ollie more difficult to set up.
Pushing switch is not considered mongo. But does look like mongo.
In many European languages such as Spanish and Portuguese, "mongo" is a slang term for mentally disabled person, in the same vein as "retard", derived form "mongoloid", an outdated medical term meant to describe the physical characteristics of those born with down's syndrome. It is likely this term started as an insult toward those riders.
It could also have its origins in Mongol, because the quick successive pushes often associated with mongo pushing makes you look like a Mongolian riding a horse.