This didn’t use to be a problem, all bars were the same size of center section, and since alloy bars and stem were invented, whatever stem you had just about all bars would fit. Now you have a choice, there is the standard size or now an oversize middle section which is 31.7, which is quite a lot bigger and more comfortable to hang onto when climbing
Of course you probably have a cycle computer and a heart rate monitor on your bars so you wont be holding on in the center anyway. Most bar and stem manufacturers make both sizes, but make sure you get the same size, its usually a good idea to stick to the same make.
All bars and stems used to be steel, now you can have alloy or carbon. Steel was strong but heavy, some track sprinters still use steel but they are the only ones that can afford to carry that kind of weight.
Width is very important, if you get bars that are too narrow your hands will be too close together which will constrict your chest and your breathing, if they are too wide then you will have difficulty controlling your bike, there is nothing worse than a small rider with wide bars struggling to sprint and the bike wobbling all over the place.
If you hold the bars on the drops your arms should be straight, or hold them up to your shoulders and they should be about the same width. Be careful when you order your handlebars as the different manufactures measure the width in different ways, some measure from outside to outside of the ends of the bars, others center to center, so what is 44 from one company is a 42 from another.
Depth is a matter of choice, but you tend to find smaller riders want shallow bars with less drop and the bigger rider with a deep chest has a deeper bar with a longer drop.
The last round: normal or anatomic?This is where personal choice comes in, try both and see what you think, if you can!
Bars were always round, and then someone thought it would be a good idea to put flat sections on the lower section of the drops, just under the brake levers.
These are “anatomic”, personally I find them uncomfortable, and you can only put your hand in certain places, with about two variations.
With round bars, you can have your hands anywhere, there are many positions for your hands, much more comfortable.

I said you should try both types, if you can, this is because round handlebars are very difficult to find in the shops, the manufacturers only let the Pro’s ride them, so for us lesser mortals we either have to put up with anatomic bars or hunt the internet and magazines.
My personal choice, at the moment, is Deda Elementi Newton shallow (Italian style) in 44 width and oversize center section (31.7) to go with my carbon oversize Time stem (120mm length).
I have just returned from a months cycling holiday in France. I noted a large number of cycles had curly handle bars that looked really comfortable – they looked almost like a figure of 8. I have searched the internet but can’t see them – any ideas what they are and where I coul;d buy some?
They tend to be called either touring or butterfly handlebars.