Router Mode controls both DHCP and NAT.
Access Point turns off both DHCP and NAT. This is also known as Bridge Mode.
DHCP Mode controls DHCP, but NAT is turned off.
if i manage a wired connection to the belkin router, what mode do i need to put it in?
Access Point. Configure the Belkin to use the exact same wireless network name, exact same wireless network security setting, and exact same wireless network password as the AirPort router.
You already know the name of the AirPort wireless network and password. The AirPort is using WPA2 Personal wireless security, unless you have changed the setting. WPA2 Personal is also known as WPA2--PSK--AES on some routers.