RAID is "Redundant Array of Inexpensive (or Independent) Disks." Basically, a mirrored RAID (or RAID 1) is a set of two disks where the information is "mirrored" or duplicated on both disks.
This requires two identical hard drives in your computer, and a "RAID controller." Fortunately, motherboards frequently come with RAID controllers built in these days. You set up the RAID when building the computer, and if a drive goes Tango-Uniform, you still have all your data (it's mirrored on the other drive). You replace the failed drive, and the RAID array rebuilds itself (copies the data from one drive to another, restoring the redundancy, and hence your data security). Mirrored RAID basically saves you from hard-drive failures.
Network Storage is a similar idea, except it's usually some form of hard-drive-containing appliance that connects to your ethernet network, and is generally accessible to all the computers on that network (access controls can be put in place to restrict access). Not only does it give you a common place to save files (and share files among different computers), but many "network storage appliances" come with built-in RAID features (so, like your RAIDed computer, your common data-storage location doesn't get taken down by a disk failure).
There are a bunch of manufacturers of network RAID appliances...
Buffalo is one, but I'm partial to
Netgear (they used to be Infrant, and make a nifty linux-based network appliance... Netgear bought them out). They're about the size of a toaster, and you just slip in a set of SATA drives, and you're off (they hold up to two Terabytes, IIRC). You can also add drives as you go, and the appliance can automatically expand to fill the space provided by the new drives.
I'm a network storage fan... I'll never go back to sneaker-net.