In general, snakes can open their mouths at about a 150 degree angle. That's pretty amazing when you consider 180 degrees is a straight line!
The reason that snakes can open their mouths so wide is that their jawbone is connected by ligaments, rather than hooking into the skull like other animals.
As you can see from the human jawbone below, the connecting points make a 'U' shape. That allows the jaw to rock back and forth (open and closed), but the movement is extremely restricted to the connection to the skull and facial muscles.
Compare this to a snake jawbone. The snake's jawbone really isn't connected to the skull in nearly the same way. Rather than a (for lack of a better term) physical connection, the jaw is connected through ligaments and tendons.
The "small bone" is connected to the back of the skull, and the mandibles are connected to that. This allows for the snake to drop its jaw further than if the jawbone was hooked into the side of the skull. It also doesn't really "break" it's jaw. Rather, the tendons and ligaments that hold it all together are fairly stretchy, allowing even more movement.
Another fun fact about snake jawbones is that the mandibles aren't fused together like human jawbones are. Human jawbones actually start as separate mandibles, with cartilage in the chin, but the cartilage fuses into bone as the person grows older.
Snake mandibles are (like the connection between the jaws) connected by tendons and ligaments, so that their bottom jaw can stretch wider if needed. Though that is used more for wider prey, or even eggs.