I don't usually see modems or routers referred to as a "bridge". I'd speculate they are talking about whether or not the device performs NAT, breaking out one WAN IP address into a subnet of 10. or 192. internally for all your devices. "half bridge" and "full bridge" may describe what is more commonly referred to as "passthrough or NAT".
In the case of passthrough, your computer (or router) is requesting via DHCP an address, and is being provided by your ISP, not your modem. (the request "passes through" the modem to the ISP) In that case the modem is acting as a transparent piece of hardware as though the ethernet cable was run from your router straight to their patch board at their office.
For non passthrough, your modem is performing NAT. Usually it's not recommended to have another router follow that, which is likely to NAT yet again. Stacking NAT causes performance problems and usually makes UPNP dynamic portmapping not work. Airports will turn their light yellow and flag a problem if they detect they are set for NAT, and their uplink is also providing NAT. A switch is usually sufficient in those cases instead of a router. Or you can just turn off NAT on your router, and it becomes passthrough, basically functioning as a switch. A router can be useful though if it's providing your wireless.
So if your modem is set to passthrough, configuring your static WAN address must be done in your router. Otherwise it needs to be done in your modem. (if your modem is passthrough and so is your router, or you use a switch instead, static must be set up in your computer)
