Basically this comes down to a question of how hands-on you like to be with your devices.
In most home-networking configurations I've had my hands on, the modem is a simple one-port device, so the router got the task of supplying network connectivity to the computers in the household. The modem did PPPoE, the router did everything else -- DHCP, NAT/Firewall, DNS, etc.
There are the odd ducks that have had nice featureful modems, and the decision was made to keep them enabled for most networking services, but add the other router as an AP to extend the wireless range. It's got some advantages, but there are lots of gotchas.
You've got the choice:
- Turn off all features of the modem (set it to Bridge mode) OR leave DHCP and PPPoE ON, but make sure the WiFi is OFF, and only connect it to the Airport. (Leaving the modem's DHCP will work because the Airport is the only thing that will connect to it.)
- Connect ethernet wire from the modem into the WAN port on the Airport Extreme.
- Configure the Airport to handle everything else (if the modem has a Bridge mode and you're using it, this includes PPPoE). If you didn't turn on Bridge mode, you need to make sure the Airport gets a WAN address through DHCP (that's from your modem), and set its LAN address to static 192.168.whatever.1.
- In this configuration, the Airport is the gateway for all your networked devices.
- Pros: one box to administer once you get everything working.
- Cons: no range benefit on the wifi from having two devices.
Complicated: most anything else.
- Leave the firewalling and PPPoE to the modem; set the modem to a static internal address (it's your gateway, so 192.168.whatever.1 is appropriate).
- Set the Airport to a static address of 192.168.whatever.2; it will be your DHCP/DNS server.
- Turn OFF the modem's DHCP and DNS functions. Turn the Airport's DHCP and DNS functions ON. Make sure its DHCP server won't give out the .1 and .2 addresses to clients (maybe set it to start at .50, or .100).
- Leave both WiFis ON, but configure them as desired (you could use encryption on the Airport for your "private" network, and leave the modem open-access for a "public" network if you like). Make sure the channels are different by 3-5.
Connect the Airport to the modem with a (long) ethernet cable, and put it ... wherever. The other side of the house, for best WiFi range everywhere in your home. Near the stereo, if you have a network-enabled PVR or HTPC and need a nearby ethernet port.
Pros: lots of flexibility here. you can adjust for range, to accommodate specific devices, whatever you like.
- Cons: you are the network administrator, and this stuff can get complicated.
Take your time, read other home-networking questions on this site and others, list out what you need from your network and map out your design. (Edit: here's a good question from someone with a network design in mind and specific goals he's trying to accomplish -- take a look.)
Remember the basics of network troubleshooting: Does machine A have a network connection (check cable or wifi status)? Does it have the expected network address (ipconfig/ifconfig)? Can it see the router (ping)? Can it see machine B (ping)? Can machine B see it (ping, from machine B)? Can it see the gateway (ping)? Does DNS work (nslookup)? Can it see the internet (http://google.com/)?