Many cyclists change the handlebars on their bikes for a number of reasons. If you are commuting or cycling recreationally, a higher rise and wider "riser" bar will give more control and a more relaxed riding position.
Mountain bikers generally will ride with flat or riser bars, and this is down to personal preference. Some people prefer a more comfortable upright position that the riser bar will give you. Alternatively, some people may like to have a lower race like position so would use a flat or shallow rise handlebar. The wider the handlebar the easier it is to turn the front wheel. Downhill type bikes generally have wide handlebars fitted to help keep the bike in a straight line over very rough terrain. For a long time, handlebars were the same width but in recent years' wider options have become available in both alloy and carbon models. Alloy bars are light can be anodised in all sorts of colours to customise you bike with. Carbon bars all tend to be dark in colour but the technical properties of the material make for an extremely light and stiff handlebar.
Road bike handlebars have variants all over them. You can have options of widths, drops, flat top or 'ergo' tops and some are swept back on the tops for comfort. Couple this with differing materials such as complex alloys, steels or carbon fibre and there really is a lot of choice for a road bike rider. Generally, the width equates to your shoulder width. This is so the position can be as comfortable as possible as your arms are in a natural position. The depth of the drops depends on your personal flexibility. How low you can lean forward and again still be comfortable. Ergo tops are usually used for aerodynamic properties. This is important for some and not others. Material comes down to personal choice, with most components that are available in carbon you usually find that they are what the professionals would use however many pros still use alloy therefore alloy bars are still the most common.