The design of a circle hook is what it sounds like. The point of the hook circles around and points back at the shaft of the hook. You can get either short shaft or long shaft hooks. With the short shaft ones the point of the hook points back just below the eye of the hook. On longer shaft hooks the point is further down the shaft. They come in many different sizes nd colors for just about any size fish you are trying to catch. Many manufacturers make them, including Gamakatsu,(compare prices) Mustad, (compare prices)Eagle Claw, (compare prices) Owner, (compare prices) and many others.
With more open, traditional styles the hook point is parallel to the shaft of the hook so the hook bites into the fish wherever it is. If the fish swallows the hook down its throat it will hook there, or in the gills if not as far down the throat. A circle hook will slide along until hit hits an exposed part of the mouth of the fish, usually the just inside the lips.
Since circle hooks usually hook the fish near the lips, it makes unhooking them much easier and means you are less likely to injure the fish when taking the hook out of the fish's mouth. That is important if you plan on releasing the fish.
Since the circle hook slides along until it hits the inside of the lips, a steady pressure on the line when the fish bites is important. If you jerk the rod tip to set the hook as is normal with other styles of hooks you will probably pull the circle hook completely out of the fish's mouth. A steady pressure allows the hook to slide along and stick in the fish.
Catfishermen use circle hooks a lot. With catfish you often put rods and reels in a rod holder and wait on the bite. When the catfish picks up the bait and starts to swim off it hooks itself. This works well with many kinds of saltwater fish, too.
With some fish, like bass in freshwater. circle hooks don't work as well. A bass will usually spit out the bait as soon as it feels anything unusual so you are not as likely to hook them. And bass fishermen usually use artificial baits where circle hooks don't work as well. When using live bait for bass, like big shiners, circle hooks will do a good job.
Other fish like freshwater stripers offer good targets for circle hooks, especially when using live bait. And if fishing for stripers with rods in holders the same techniques apply that work for catfish.
There is some evidence circle hooks result in fewer hook-ups. They are required in some saltwater tournaments and studies like the one from North Carolina State University show a lower hook up percentage per strike than when using traditional hooks.
If you fish in ways where circle hooks work, and want to release some of the fish you catch, give them a try.