In case you don't know what a Delayed Write Failure is, a brief explanation. All modern operating systems cache data writes to improve performance. Hard drives are extremely slow, when compared to RAM access (which is slow when compared to L2 Cache, and so on right to the CPU's registers). The OS assumes that most of the time when you use data on the drive and save it, there's a fair possibility that you might save that data again some time soon (or a nearby, similar piece). It saves time to just store that data in RAM for a few moments (and act like it saved it to disk to you and the rest of the programs running) and "save up" until you have a big enough batch of data to write all at once. Also, if you re-change that data (ever click save two or three times in a row on a Word document?), it doesn't waste time writing out to the slow disk the first time(s) by using the exponentially faster ram. This is called a delayed write. The OS is holding, or caching, that data until it's good and ready to write it (which usually happens within a few seconds).
A delayed write failure is when later comes, and the system tries to write the data out to the disk, and something doesn't work, so the data gets lost (or partially lost or maybe it got there but didn't confirm that it did, the OS doesn't know). This isn't good, because the data lost could be only part of a file, or there could be random data overwritten in the middle of other files (if a file only got 1/2 moved). Either way, bad news. The write shouldn't have failed unless you disconnected the drive or shut down the computer -- which is why you shouldn't unplug external drives without "disconnecting" them in the OS first and is why you should always "shut down" Windows properly and not just yank the plug.
Assuming you weren't doing those things... This can also be a very bad sign for the drive's health physically (or the drive's cable). Obviously, if it randomly can't write data, something is wrong. I would probably take immediate action to backup any data on the drive. Remember, its never a question of "if" a drive will die, its a question of when. We haven't yet figured out how to build a machine that spins at 7200 RPM 24 hours a day, 7 days a week, and have it last forever -- not if we plan to sell it on Newegg.com for 50 cents a GB anyway. Eventually, on the 20 billionth spin round, something in that mechanism is going to break or wear down.
So... Lecture complete. Back it up, then check the disk format for errors. It could be a 100 things wrong in software, but most of them are unlikely if you haven't been mucking around in the registry, overclocking, or running weird drivers. Also, swap out the drive cable (either IDE ribbon, USB, Firewire, or SATA ... whatever, just try it with another cable), as that is a common culprit. Even if swapping the cable (or investigating a sofware fix) proves to be a culprit, I would probably plan to reformat the drive anyway and restore from backup. All that random data lying around on the drive can a be little like digital "rot" (as some people call it) At one point when my old media drive was dying (and throwing delayed write failure errors) I ended up with a bunch of corrupted music files (weird things, hunks of static thrown in, bits and pieces of songs mixed up in others, and a pile of songs that would crash explorer if you simply selected them).
Either way, something to address immediately.
EDIT: I hadn't remembered to mention when I got into this rant that you referred to network drives. This could also certainly be a network issue (for the same reasons, something blocked the data from being properly saved -- it could be any of the devices "in the way"). You'll really need to check all of the links in the chain. An easy way to do this is to try to eliminate them. Can you reproduce errors on the system locally (without using the network at all)? If so, then look to the drives, if the network seems to be required to cause errors check the switch, NICs, and drivers. Look for things that have changed and try to reduce complexity.