In the context of bottled beer, the "dregs" refer to the cake of (mostly) yeast and coagulated proteins that form in the bottom of bottle conditioned beers.
The reason why anyone would care about this gunk (which you usually don't drink), is that since its mostly yeast, and mostly still viable/alive, you can collect these dregs and add sugars to grow them up into a viable amount for pitching into a batch of your own beer.
With most 'normal' bottle conditioned commercial beers (Belgians, some Englishes, etc), caring about the dregs is less interesting than with the sours. Basically, the yeast at the bottom of a Chimay bottle is virtually the same as a strain of White Labs (WLP 530, i think?) that I can buy for $6 down at the homebrew shop. However, if you don't have a local shop, or if you're just a yeast nerd who likes the culturing process, you can use those dregs to build up sufficient yeast for pitching into your own beer.
With Sour Beers, the blend of yeasts, Brett, Pedio, Lacto, etc that's found in their bottle dregs is something much more special. It used to be quite difficult to get ready-to-pitch Sour Beer yeast cultures, as there wasn't much demand for them at the homebrew level going back maybe 10 years ago. Now, with sours being in vogue, you CAN get some strains, but that's besides the point. Back then, your best choice for cloning something like Orval, for example, would be to pour the dregs of a bottle of Orval into a starter, or straight into a fermented beer.
Extra info: Not all dregs are created equal. Some breweries pasteurize their beer, then add a different strain to bottle condition. If you are going to use dregs, google the brewery first to ensure that they bottle their beer with their primary strain. For example, some Hefeweizens are bottled with lager yeast, so if you cultured those dregs looking for Hefe yeast, you would not get the results you're after.
It means exactly the same thing in brewing...liquid remnants.
Expanding a little, typically the yeast sediment you see at the bottom of the bottle of your home brew is referred to as "the dregs".
As a home brewer becomes more experienced and familiar with yeast, they may actually use a technique called "yeast harvesting," where they actually harvest "the dregs" from the bottom of a bottle to re-create a beer they want to brew again, or even clone one of their favorite microbrews (some commercial microbrews still have 'the dregs' at the bottom of the bottle, and can be harvested).
Resource
"Yeast Harvesting / Re-Pitching" via wyeastlab.com
it is the grains, or yeast etc that used to be floaters that settle. so, Like Denny said, it is the liquid left at the bottom of your fermenter, or bottles, or even kettle, with the sediment.
