NAS stands for Network-Attached Storage. Basically a power-efficient computer or server that lives on your network that you use for keeping stuff. Makes it easy for multiple people to access the same files, so a great place to keep movies and music.
I currently have a QNAP TS-231P, but it functions both as a NAS, and as a server running some useful software (see the linked page for more info.). I highly recommend, if you're into messing with Linux or BSD and computers, building your own. I certainly wish I'd done that, as so-called easy solutions like QNAPs and Synologies have a bunch of shit you don't need, and aren't great at having stuff up-to-date.
There's a popular NAS-targetted OS called TrueNAS (previously FreeNAS) that I'm curious to try. But I'd also like to just run a server with regular Linux or BSD maybe?