Illegitimi non carborundum
"Don't let the bastards grind you down"
This is a joke Latin phrase; or rather, it is a mock-Latin phrase, as it has no real connection to Latin at all. It's hard to say why it has been so popular for so many years, but it is a bit of nerdy punning that has been around for decades. Sadly, when new generations discover it they often find themselves critically confused -- probably a combination of the decline in knowledge of Latin among the general population and the intense feeling we all have that, surely, the memes of our grandparents couldn't possibly be as dumb as Nyan Cat or The Llama song.
Carborundum is a brand name for a manufacturer of silicon carbide, an abrasive. In the early 1900s it was common to see advertisements for Carborundum, usually as an easy and cheap way of sharpening razors and knives. As such, it was immediately and unquestionably recognized as 'not Latin'. The "illegitimi non" part is more-or-less-Latin. Not grammatically correct, and not intended to be, but recognizable as Latinish.
One important note, that may make the attraction of this phrase more evident: there is no English phrase "don't let the bastards get you down". Or rather, there wasn't. The original form seems to be Illegitimi non carborundum, which has been corrupted in a number of different formulations, a few of them being English translations. This was a newly created way to jokingly call your competitors bastards -- without actually using a bad word, or even directly referring to one. If someone wanted to try and translate an obviously nonsensical phrase, and if they were to come up with a translation involving such a rough word as 'bastard', well, that wasn't your fault.
It is generally said that this phrase first appeared during World War II, where it was a popular catchphrase among the troops; however there are reports of it appearing on witty wall-hangings even before then. Regardless, it was certainly popularized by US Army general "Vinegar Joe" Stillwell appropriating it as his motto during the war. It was brought even more fully into the public eye when the 1964 Republican presidential nominee, Senator Barry Goldwater, took a shine to it; he is said to have had a sign with "Noli Permittere Illegitimi Carborundum" on it hanging in his office.
If you seek to dig deeper into the roots of this phrase, be warned! It is a common pastime of history and language nerds to make up and post fake etymologies of this phrase. These are done with little indication that they are jokes, and many of them have been posted repeatedly across teh internets by people with little interest in researching their factoids. Anything that tries to tie this phrase in with an actual Latin word 'carborundum' is a joke.
Of course, the whole thing is a joke, so that's okay. I will leave you with one of the most famous instances of Illegitimi non carborundum, Harvard University's most popular fight song, Ten Thousand Men of Harvard; this song was originally published as Harvard Holds Sway (words by Alfred Putnam and music by Murray Taylor, Harvard College Class of 1918; published in 1914). It is unclear when the Illegitimi non carborundum verse was added, but it was apparently not yet part of the fight song tradition in 1975. It has since become an integral part of the song:
Illegitimum Non Carborundum;
Domine salvum fac.
Illegitimum Non Carborundum;
Domine salvum fac.
Gaudeamus igitur!
Veritas non sequitur?
Illegitimum non Carborundum -- ipso facto!