My Windows 2000 machine is connected to the internet. I'd like to share the internet through this machine to the Slackware 10.1 system.

W2K system has two NICs. The one connected to the internet has the ISP-assigned IP address, the other is set to 192.168.0.1 w/ netmask 255.255.255.0. Slack system has the NIC configured with an IP of 192.168.0.2 w/ netmask 255.255.255.0.

W2K system has internet access. Slack system can ping 192.168.0.1 with positive results, but cannot ping any other IP address or domain name.

How must eth0 be configured so that the Slack machine can access the internet?

Your problem's on the Win2K side; it needs to be configured to share it's internet connection. I don't immediately remember how to do that in W2K, but poke around in Control Panel->Network Connections->Properties of internet-connected NIC. It SHOULD be fairly straightforward, but don't quote me on that.

Edit:
Here's a link to get you started. Google is your friend.

W2K machine is already setup to be shared, I used that exact guide as a matter of fact.

Anybody else got some advice?

Configure networking to use DHCP.
or for a static IP address
Assign gateway IP adress to 192.168.0.1
Add nameserver IP address information to the resolv.conf file in the format
nameserver xxx.xxx.xxx.xxx