(Probably more than you ever wanted to know.)
Non-volatile memory is memory that doesn't go away when you remove power, like a hard drive, SD card or SIM card. RAM loses everything in it when you turn off the power - it's volatile.
SIM cards carry data in them - in a non-volatile memory chip in the card - the same as an SD card. You can keep contacts on them (you should keep them in Google Contacts, so you can easily sync them to a new phone, but you can keep them on the SIM card) and they don't disappear when you take the card out of the phone, which makes the memory non-volatile.
(Cards don't "carry" memory "on them" any more than hard drives do. They store data in the memory that they are - magnetic for hard drives, electronic for cards.)