I've been using linux for a few years, but I still feel very much like a newbie. I've searched google/linux, forums, and picked some brains here at work, but I just can't get a solid explanation.

Can someone explain exactly what snapshot is for, what it does, how it works, and what happens when it breaks?

Here at work I'm told that whenever someone deletes a file from the NAS it goes to the snapshot, like the recycling bin in windows. If the snapshot fills, then it'll start eating space from the rest of the drive, and it could even break the machine. 4 times a day our snapshot cycles to a new one, but each snapshot is nearly the same - why are files 'backed up' or 'deleted to' the snapshot in multiple locations?
My research on-line tells me it's more of a backup [ghost] image, but that only raises further questions.

I can't make sense of the pieces and/or fit them together!

A snapshot is like a snapshot, hence the name. That is, it is a record of what is in the filesystem at one point in time.

Now, the implications of this depend somewhat on the technology. Conventionally, you would think that if you took 5 snapshots (over a period of time) of a 100G filesystem (100G +/- a small amount for changes) that would need another 500G of storage space and would cause a lot of traffic as that data gets copied around.

However, if you use the CoW mechanism (copy on write) to provide the snapshot you don't save extra copies of the data, you have a base data set and (effectively) a set of deltas, and you can still reconstruct the data set at any of the snapshot instants. And, perhaps as important, because you aren't physically copying data around, you aren't causing the extra traffic.

Filesystems like ZFS and the forthcoming BTRFS use CoW and are thus able to do snapshotting particularly efficiently. My guess is that you probably don't have one of those, and are probably doing snapshotting the 'brute force and ignorance' way.