Depending on the software, a dual-port network card (regardless of the physical medium it supports, either fiber-optic or copper) can usually act in two ways
1) As two distinct interfaces that can exist on separate subnets (which implies separate routing statements, etc. and in this case, a default-gateway has to be picked).
2. As a 'teamed pair' where both interfaces represent the same IP address at Layer-3. Compaq uses "NIC teaming" where there is an active and a standby interface if I recall correctly. This provides redundancy and fault-tolerance, but not increased capacity. Sun uses IPMP, which involves a 'floating MAC address' and the driver monitors each interface to maintain who owns the MAC (ethernet, or L2) address.
For increased capacity, 802.3ad (Cisco calls it "Etherchannel") is used. This method "bonds" the two interfaces together under the appearance of a single virtual interface and the driver then load-balances the traffic across the two interfacs. This method requires matching configuration on the switch that the NICs are connected to (both interfaces must be connected to the same switch) because a negotiation protocol (either LACP or LAgP) is used to manage the membership of links in the channel.
As for your question about 'dual fiber' cards, I assume you're still talking Ethernet and not Fiber-Channel which is a different animal. If my assumption is correct, then the physical medium is usually abstracted from such teaming/pairing configurations by the drivers, so whether you're using fiber or copper is irrelavant.