Presently, I have router between my modem and my Airport. That configuration had been necessary because there were no ethernet ports on my (old) Graphite Airport.
I assume your olf airport looked like
this ?
If you were using that with a cable modem or other internet source that had only one LAN ethernet jack on it, you'd need some way to get more ethernet ports to use. Usually you'd go with a switch, but if your cable or dsl modem wasn't intended to be generously handing out many ip addresses, (some are set to only give ONE) then yes you would require a router to share the one IP address it gave you with several computers using invalid LAN ip addresses.
If you've replaced it with a newer airport that has enough LAN jacks, then you should be able to just plug the airport WAN jack directly into the modem, and remove the router entirely. (you'll probably need to reboot the modem when you do this) The airport's LAN jacks are tied into the same internal router in the airport and will by default share your one IP address over invalid LAN ip's for both the wired and wireless clients.
