Stage 1: Get a Connection
I'd first try to configure your nic via DHCP:
sudo dhclient
After that, I'd expect your machine to have an IP similar to your other machine, 10.1.24.XY
You can check what your router IP is by running
route -n | grep UG
I wager your default gateway (router) is at 10.1.24.1 - ping it to verify it's working okay.
If dhclient didn't get you an IP, manually set one:
sudo ifconfig eth0 10.1.24.200 netmask 255.255.255.0 broadcast 255.255.255.0
sudo route add default gw 10.1.24.1
I hope that gives you a good feel.
Stage 2: Make it persist
I'd wager you should just goof with your network manager settings.
If you want to manually configure your nic, edit /etc/network/interfaces
Most users will want to edit their persistent network settings via the network manager: https://help.ubuntu.com/community/NetworkManager
nm-tool
? Also, are you using DHCP? Cable or wireless?