The most basic form of pedal uses toe clips or straps to keep your foot secure on a flat pedal. These are fine for occasional use as they are simple and you can wear non-cycling footwear however getting on of toe clips can be tricky.
If you ride regularly and want better performance it is worth trying out clipless pedals. Instead of a toe strap these use similar technology to ski bindings where a cleat attached to the sole of your shoe clips directly into the pedal. This makes a much more secure contact between rider and bike, you will feel more stable and you can maximise your pedalling power so none of your energy is wasted.
Clipless pedals come in a wide variety of design. Road pedals are designed to be light weight, most are single sided so you need the pedal the correct way up to clip-in and are created for riders who want the ultimate in performance.
Road shoes are also light weight, stiff for the best transfer of leg power to the pedal and have a smooth sole as you are not expected to walk anywhere. The cleats for road pedals side on the outside of the sole so can make balance and walking difficult off the bike. If you ride on the road but need to be able to walk because you are commuting or simply want to get off and explore during your ride there are shoes with more rugged soles and recessed cleats that allow you to do this. Some shoes are designed to look more like sports shoes or casual wear, perfect if you are popping into town or pedalling to the local pub.
Off-road pedals are slightly different in design although they work in the same way. Pedals are dual-sided so it's quick and easy to engage the cleat if you have unclipped to dab a corner. Mud shedding is an important consideration so open sided designs that don't hold mud and debris are common. Mountain bike shoes have more aggressive soles for grip if you have to get off your bike and cleats are recessed so as not to interfere with walking off the bike or collect too much dirt. Some mountain bike shoes are designed for racing so are stiff and lightweight, others subscribe to a more relaxed leisure look.
Not all off-road riders want to be securely clipped to their bikes. Downhill mountain bikers, dirt jumpers and BMX riders prefer flat pedals so they can take their feet of the pedals easily. Flat pedals have pins or jagged platforms that help the sole of your shoe stick to the pedal. Flat shoes have soft tacky soles for grip and the style is more casual and fashion conscious than performance orientated.