In simple terms, a NAS is kind of like a "Barn for your data". If your PC is your house, then a NAS would be a shed, or a barn. You put your stuff in it, and it's separate from your PC.
NAS units have various numbers of bay, starting with one (ok, there are "NAS adapters", for USB external HDDs, that have zero internal bays), up to whatever number, in increasing amounts of cost.
Generally, most home users use 2- or 4-bay NAS units, which allows you to run RAID on the NAS. RAID is a way of using more than one HDD, to achieve additional speed, or reliability through redundancy, as needed.
I have several NAS units. One Seagate GoFlex Home unit that I got cheap from BestBuy Outlet, and a QNAP 2-bay unit with a pair of Seagate 1TB HDDs in a mirror.
I have a Lenovo 2-bay NAS unit that I haven't assembled yet, I'm debating whether to use two 3TB Toshiba drives (which are on the hardware list for the NAS), or a pair of HGST 4-6TB NAS drives.
A "mirror" is basically a live clone of the data. It gets written to both drives, so if one drive fails, hopefully the other one doesn't go at the same time, and you'll always have one copy of your data left.
Larger NAS units, with 4 or more bays, may use RAID 5 or 6. That's a type of striping, that allows for N+1 redundancy (RAID 5) or N+2 redundancy (RAID 6). That means that you get N drives worth of storage, and 1 or 2 drives can fail, and your data will still be intact.
Edit: I should note, that I don't have any particular preference for Seagate drives. Just that the GoFlex Home came with the drive pre-installed, and the QNAP, I had the pair of brand-new Seagate drives laying around because I got a good deal on them, and I didn't have a pair of WD drives. I actually slightly prefer WD for reliability (but not Greens), but they're more expensive, and like I said, I didn't have them laying around like the Seagates.