But why "network attached". Why not local storage?
Local storage is only (immediately) available to the one computer it's connected to. Network attached storage is available to any computer within the network. If you have a desktop PC in your office, an HTPC in the living room, an iPad in the guest room and your kid(s) has a laptop, then a NAS will make (selected) files available to all these computers. That is especially handy for stuff like media libraries. Everyone can still have private folders for private stuff though.
If the NAS has more than one drive in a RAID-setup, all that data is kept redundantly. Backups (to external drives or tapes) can be made at one centralized location. As soon as you're dealing with two computers, NAS begins to make a whole lot of sense.
If you only have one computer, NAS doesn't offer a lot of benefits. From a SPCR point of view, they offer the possibility to locate the drives, usually the noisiest part in a quiet computer, away from the workplace, in a closet or so.
Any PC can act as a NAS server. But since they are ideally running 24/7, most people choose to buy or build a dedicated server with low energy consumption. A lot of manufacturers offer simple 2-drive NAS solutions (I'd always go for at least 2 drives in a RAID-1 config) at affordable prices that won't use more than 5w when they're idling and have spun down the disks.