The whole process is reportedly quite simple. After pointing your iOS device to JailbreakMe.com, hit the big "slide to jailbreak" button on the front page (which mimics Apple's slide-to-unlock button) and within minutes the hack will finish installing itself and dump the Cydia Store app on your home screen. Apple can't have the site taken down now that jailbreaking has been legalized, but the company can -- and likely will -- close the Safari hole that makes this hack possible in a future update, if only for security purposes.
Early reports said that some iPhone 4 owners who used JailbreakMe to hack their phones were having problems with FaceTime and MMS, but those problems are now fixed. Nevertheless, you should backup your device before proceeding with the hack in case something goes wrong, and of course, be aware that jailbreaking voids your warranty with Apple.