You need to know the IP address schema of your network. First of all, do you have a DHCP server on your LAN, or is each machine configured with a fixed IP?
If you have a DHCP server, simply configure your ethernet card to use a DHCP provided address and to get the DNS info from that server. (Using the Gnome menu, start "System Settings -> Network". Then edit your ethernet device's configuration. Select "Automatically obtain IP address settings with: dhcp" and "Automatically obtain DNS information from provider".)
If each machine has its own fixed IP address, instead of using dhcp you must set the "Statically set IP address" option and give your PC an IP inside your LAN's IP address pool. Ask whoever administers the network for that IP. And to reach the Internet via the router (it doesn't matter if it's DSL, it is a device on the LAN, it has an IP), set the "Default gateway address" to the router's IP.
Then restart the network entering "/sbin/service network restart" on the bash shell (as root), and you shall be able to reach both your network and the Internet. Voilá!
(I'm assuming you already have your PC connected to the switch...)