While developing I use Chrome, the dev tools out of the box are just amazing. I highly, highly recommend you give this (free) Code School course a try. It’ll interactively walk you through everything you can do with Chrome Dev tools: http://discover-devtools.codeschool.com/
Generally you have 3-4 major browser engines: Blink (Chrome, Opera), Trident (IE), Gecko (Firefox), Safari (Webkit…Blink is actually a fork of Webkit). Like felgall metioned they each run a bit differently and may even use different versions of JavaScript depending on the engine/version.
Unless I’m doing some hardcore JavaScript stuff though I usually feel comfortable just testing in Chrome and IE.