Handlebars change depending on what sort of riding you are doing. Road bike handlebars are commonly known as drop bars. They vary in shape and width which changes with the size of the rider (shoulder width should be approximately the same as handle bars). The material also varies, which is often aligned with handlebar weight. Aluminium is the most commonly used material and in higher end handlebars carbon fibre is used.
Aero bars are designed for use on triathlon or timetrial bikes. They're optimised to put the rider in a low and narrow aerodynamic position.
Flat bars are used on a variety of bikes. The majority of mountain bikes use flat bars as do flat bar road bikes. Like road drop bars they vary in material, shape and width. Alloy and carbon are the most commonly used materials.
Riser bars are much like flat bars but are designed to raise the hand position higher. They too feature commonly on mountain bikes.
BMX handlebars have a design which differs from road and mountain bikes, BMX handle bars are reinforced by having a brace bar which gives them increased strength and rigidity. This is necessary because BMX riders put a lot of pressure on their handlebars while performing tricks/stunts or dirt/race riding. Aluminium or steel are common materials here.