I sometimes see woodworkers talking about a zero-clearance insert for their table- or band-saws.
What is it? And why/when should I use one?
I can't think of a time when you wouldn't be better off using one. However, sometimes it's sort of a pain to make one for every blade, or in the case of a Dado set every blade combination... So it tends to not be used for non critical cuts sometimes. But again, I can't imagine it not helping for any cut.
A zero clearance insert is an insert which exactly matches the width of the blade. You can make one by slowly raising the blade through an un-cut insert.
The advantage is as aaron says, they prevent things from falling through the hole, and more importantly, they reduce chip-out by ensuring that the piece has support and thus chips won't tend to get pulled away from the board. It's not perfect though, so if you care, still put in your sacrificial piece. In general, the less open space you have under your piece, the more control you have over it when cutting.
You should always use an insert that leaves as little gap around the blade as possible.
Let me add a couple things:
They're typically made out of plywood or MDF to prevent wood movement.
To add to the dust collection argument. I own an Atlas table saw made way back in 1938. They cared so little about stopping dust back then that the stock insert threw more than half the dust in my face, on the floor and onto everything in my garage. A quick vacuum volume test showed around 30% made it into the catch bin under the saw, rest above the saw. The stock insert was so horrendously over sized for a standard 10" saw blade that even cutting a 1/4" vernier slat took many attempts to get right.
After producing my first zero-clearance insert. So little dust got out through the top that I can sweep it up with a paint brush after an entire day of use. It began to fill the catch bin so rapidly that I had to stop all other projects to work out a proper dust collection system for that saw (poor Tidy Cat container, it was once a bucket, now it's a clipboard).
Two points: The only place I (first) heard of ZCI is when building and using my router table. Router blades benefit greatly by there use. They prevent much tear out from elaborate shapes ripping out wood both with and across the grain.
The other small point is that the second purpose of woodworking is the creation of sawdust. If its careful collection becomes your main goal you may as well use your garage for cars and bikes.