Illegitimi non carborundum is a mock-Latin aphorism meaning "Don't let the bastards grind you down".
The word illegitimi (i.e., the plural of illegitimus, actually translates to "unlawful" or "outlaw" but resembles the English "illegitimate") is sometimes given in the plural dative case (that is, illegitimīs), a case that follows the Latin gerundive and denotes agency. Non negates activity. Carborundum is an industrial abrasive material also known as silicon carbide, but its name resembles a Latin gerundive, a grammatical construct that expresses desirability of whatever the verb denotes. So carborundum can be mock-translated to mean "fit to be ground". Thus "illegitimi[s] non carborundum" would mean "not fit to be ground by the illegitimate", or "Don't let the bastards grind you down".
The phrase originated during World War II. Lexicographer Eric Partridge attributes it to British army intelligence very early in the war (using the plural dative/ablative illegitimis). The phrase was adopted by US Army General "Vinegar" Joe Stilwell as his motto during the war. It was later further popularized in the US by 1964 presidential candidate Barry Goldwater.
The phrase is also used as the first line of one of the extra cod Latin verses added in 1953 to an unofficial school song at Harvard University, "Ten Thousand Men of Harvard". This most frequently played fight song of the Harvard University Band, is, to some extent, a parody of more solemn school songs like "Fair Harvard thy Sons to your Jubilee Throng". The first verse goes:
The phrase is also used as part of a student painted crest on the bottom floor of Hodge Hall at Princeton Theological Seminary.
A wooden plaque bearing the phrase sat prominently on the desk of former Speaker of the U.S. House of Representatives John Boehner.