"Non illigitimus carborundum" is not proper Latin. Rather, it uses the principles of Latin grammar to make a play on words. A joke for language or grammar lovers, it means "Don't let the bastards grind you down."
"Non illigitimus carborundum" was made up to to sound like Latin, but it isn't. It only mimics Latin word endings. Anyone with a little knowledge of Latin or word meanings will see the joke.
"Illigitimus" looks as if it could be the Latin for "illegitimate," or "bastard." (The actual Latin word for this is "nothos.") "Carborundum" is also not a Latin word, but the name of a hard stone used for grinding. The ending "-undum" suggests the Latin gerundive form, which is used to express actions that must or must not be performed.
The phrase seems to date to British army intelligence during World War II. It spread from there, and has been adopted into a line of the unofficial Harvard University school song, and as the motto of organizations including the University of Idaho Navy ROTC Drill Team and the Laughing Dog Bicycles bike shop in Amherst, Massachusetts.