A bivvy sack is what they call a bivouac sack for short. A bivouac is a kind of temporary camp that soldiers use generally, where they don't really use tents or anything. They just roll out their sleeping bags and sleep right there, wherever they are. Mountaineers like the bivouac concept too. When they climb tall or difficult mountains, often, they have to take a couple of days doing it. Mountain slopes are kind of hard to the pitch camp on, and they just use a bivouac sack to sleep wherever they are.
As you've probably noticed, we still aren't really talking about what exactly a bivouac sack is. Is it like a sleeping bag? Not quite. A bivvy sack is a kind of waterproof and weatherproof cover that you pull on top of your sleeping bag. If you need to sleep out in the open without the shelter that a regular tend provides, you wouldn't want the rain or dew to drench you. You use this cover for your sleeping bag, and you are quite assuredly dry.
Early bivvy sack designs were nothing more than waterproof nylon bags that you the zipped yourself up in when you were in your sleeping bag. Modern day the designs though improve on the original concept. When you have a regular waterproof nylon cover for your sleeping bag, the waterproofing works both ways. It won't let vapor from sweat escape, and it won't let water from outside in either. This doesn't make for a very comfortable sleeping experience.
With the modern bivvy sack, you get a dual layer design. The inner layer is nylon and urethane. The outer layer is Gore-Tex. You get breathability this way. The modern bivvy sack does two things very well it keeps your sleeping bag dry, and it makes it warmer inside by as much as 10 degrees.
Manufacturers since have thought up many improvements to the basic design. Some models offer armholes so that you can strap yourself up inside nice and warm, and still continue to do work outside. Some of them offer you a full-length zipper instead of just the hole on top of to enter and come out of. It's just a better way.
When you set out to buy a bivvy sack for yourself, try to consider how much breathability the fabric actually gives you. Look for actual genuine Gore-Tex for the best effect. It's often a good idea to make sure that the bivvy sack you get isn't too snug a fit. It can make for an uncomfortable night to be restricted in this way.