Road handlebars are mostly made in alloy and have a nice bend to them for comfort of your hand on the drops, but some have a flat section just below where the brakes go and depending on your hands these can be uncomfortable for your wrists, also available are oversize bars.
These have a fatter section at the stem, are stronger and can be more comfortable. There are more carbon handlebars these days, but most of them are made with the bar and stem in one piece and you can not adjust the angle of the bars, these are also more expensive, but do flex more than alloy.
Mountain bike handlebars can be straight or with a slight bend in them, these are best for racing and with the use of bar-ends can be comfortable with more different hand positions and are available in alloy or carbon. If you want a more relaxed position there are available now a more bent bar, the bends take more of the bumps out of your ride and give you a more upright position, these do look like the bars from a shopping bike, but are comfortable.