Let's start by getting the obvious one out of the way.
When exposed to the glowing green chunks of his former home planet, Superman is greatly weakened and long-term exposure can even result in his death. It is Kal-El's greatest weakness, that and the fact that the universe seems to hate him enough to make sure every single lump of kryptonite winds up on Earth.
Over the years, it has been used by everyone from Metallo (who uses it to power his robot body), to the Kryptonite Man. Even Batman keeps a sliver of the stuff, just in case Superman were to ever go rogue.
Most famously though, it is used by Lex Luthor to power his Anti-Superman Battle Armour. He also used to wear a ring made of kryptonite, but that had the slightly inconvenient side-effect of giving him cancer.
There are also various other forms of kryptonite for when writers are feeling lazy. There's red kryptonite which causes Kryptonians to lose all inhibitions, gold kryptonite which can completely remove Superman's powers and a whole host of other weird and wacky variations. One Silver Age story even featured red-green kryptonite, which inexplicably caused Superman to grow an eye in the back of his head.
I was just a mild-mannered NCTJ accredited journalist until one day I found out the truth... that I could share my nerdy ramblings with people on the internet! It's just like mumbling to myself on the train, but without all the strange looks.