A Bunny Hop is the act of making a bike jump into the air. Bunny Hops originated as a trick done on a BMX, but have become an essential part of any mountain biker or urban cyclist's skillset, as they provide an easy way to avoid rocks or curbs. Learning how to bunnyhop is a long process. Being confident enough to do it at speed whilst dodging a bus will take longer still.
It helps to start with either top-clips or clipless pedals (the ones where a cleat on your shoe locks into the pedal) as this lets you familiarise yourself with the weight distribution without needing to worry about the big issue of lifting the rear of the bike. However, even if you regularly use clips, learning to bunny hop on flat pedals will improve your technique and confidence. As with any bike trick, somewhere quiet and flat is required for the learning process, a grass field is ideal.
In order to bunny hop, you must be travelling reasonably fast, as this makes the balancing easier. Travelling too fast, especially at first, is a bad idea, for obvious reasons. Once you are moving, stop pedalling, and rotate your pedals so that the crank arms are parallel to the ground. The choice of which foot is at the front differs from person to person, but you will find one is much more comfortable than another. Pull up on the handlebars, and shift your weight back slightly, as if you were going into a small wheelie. Once the front wheel is about a foot into the air, shift your weight forward slightly, and in the same motion pull up on the pedals. With the help of the toe clips, this should be fairly easy, and you should find yourself airbone. Practise, and you'll get the hang of it within a week.
The hardest bit is learning how to pull up on the pedals without anything holding your feet in place. This seems like a physical impossibility, as any upward motion with your feet should cause them to leave the pedals, most likely causing you to loose your balance and crash. The trick is to tilt the pedals forward, at an angle of roughly 45 degrees, and simultaneously push backwards and pull upwards. The backwards force keeps you feet locked to the pedals and provides enough friction to allow the upward force to lift the bike into the air. Pushing forward and up on the handlebars will help the bike gain more height at the same time. Learning this is not easy, and you can expect to fall off several times, before finally getting maybe half an inch into the air.
Once you have learnt the basic bunny hop, you will need to practise lots to gain height. Once you can comfortably jump half a foot into the air (which will take a few months), you can start to practise jumping up curbs. The easiest way to learn how to do this is to just ride at a small curb, hop, and hope for the best. You are unlikely to do any damage to yourself or the bike if you hit the curb. Build up your speed and height until you can confidently land both wheels on the curb simultaneously, then you should be confident about clearing other obstacles.
Fun things to practise jumping over include park benches, rocks, bricks, logs and your friends. Having a bike land on your abdomen doesn't hurt as much as you might think.