How to access data from other computers when I am connected across the internet

The normal way of doing this would be by using a VPN (Virtual Private Network) solution.
This enables you to access a remote system as if you where on the local LAN, some slight lag due to encryption, but with all the same rights and permissions.

In the home environment you can make use of Remote desktop, built in to the likes of MSN messenger.

What are you trying to do ?

Another tool is one of the services like logmein.com or gotomypc.com. Logmein offers a free service, which doesn't include file transfers. But you can then e-mail the file to an address you have access to from where you are.