In their rooms, hotels can offer either wireless or wired high-speed connections (or even both).
For wired, you would plug the hotel's ethernet cable into the ethernet port of your laptop. You would open your web browser, and you would most likely immediately receive some type of "welcome" screen. If the high-speed service is on a paid basis, you would need to acknowledge that you accept the charges (which would appear on your bill when you check out). Once past this welcome screen, you are on the "open" Internet. You would access your Earthlink mail and functions via the Earthlink web pages that Earthlink specifically provides to its customers for this very purpose.
For wireless, it's essentially the same, except that there's no cable. Depending on your operating system and your computer's setup, you might need to enter the SSID (the identifier for the hotel's wireless network) or (as my laptop does) it will find it automatically - you will receive the same type of welcome screen as described above, and once you're on the open Internet, you'd access Earthlink in the same manner.
The bottom line is that at no time would you need to actually use Earthlink's proprietary network, and no dialing up of any kind would be required.
For overseas locations, there should be no difference. I've used both wired and wireless highspeed in Germany, Austria and Italy within the last few months without any problems whatsoever.