Bicycle shoes offer several advantages when you’re riding, starting with the more rigid soles. They’re almost always designed to be used as part of a clipless pedal system, but even without the rest of the system, the rigidity of the sole means more power transfers from the whole of your foot to the pedal: making it easier to go faster.
Where cycling shoes really shine, though, is when you clip in. A metal or plastic cleat attaches to the sole of the shoe beneath the ball of the foot, and then the cleat clips in to the pedal itself. When you’re connected to the pedal, every part of your pedalstroke contributes to going forward. On a regular flat pedal, the 30% at the bottom front of your pedalstroke is what sends you forward. With a clip system, it’s more like 70%. Ride longer and train better with more efficient pedaling.
There are two general styles of cleat: road, or three-bolt and mountain, or two-bolt. Road cleats are generally larger, sit above the surface of the shoe significantly, and are almost always plastic. The large cleat shape lets you get a better connection to the pedal. Mountain cleats are generally smaller, sit flush with the tread of the shoe, and are almost always metal. The recessing gives you surface to walk on, and, combined with the metal cleat makes for a more durable system.
Which system is a better fit for you depends on a number of factors, starting with which style of riding you want to do. Let us know if you have any questions!