Most OEM handlebar tape is some kind of synthetic cork. There's all kinds of replacement types, styles, colors, Etc. I personally like tape that's a grippy or tacky type. Obviously you can buy it cheaper online, but it's good to go by your LBS and see what's available, as there are lots of options these days. They can even tape it for you if you're unsure how to make it look nice. If you're a heavy sweater, it's a good idea to install new tape frequently. I've seen more than a few handlebars broken from sweat corrosion. Every year or two is probably good for the average bike rider and it's a good time to replace the cables as well.
I made a video several years ago of the method I like to use for wrapping bars, which is pretty much how I still do it.