When a client connects via bridging to a remote network, it is assigned an IP address that is part of the remote physical ethernet subnet and is then able to interact with other machines on the remote subnet as if it were connected locally. Bridging setups require a special OS-specific tool to bridge a physical ethernet adapter with a virtual TAP style device. On Linux, for example, brctlis this tool. On Windows XP or higher, select your TAP-Win32 adapter and your ethernet adapter in Control Panel -> Network Connections, then right click and select Bridge Connections.
When a client connects via routing, it uses its own separate subnet, and routes are set up on both the client machine and remote gateway so that data packets will seamlessly traverse the VPN. The "client" is not necessarily a single machine. It could be a subnet of several machines.
Bridging and routing are functionally very similar, with the major difference being that a routed VPN will not pass IP broadcasts while a bridged VPN will.
When you are bridging, you must always use --dev tap on both ends of the connection. If you are routing you can use either --dev tap or --dev tun, but you must use the same on both ends of the connection. --dev tun tends to be slightly more efficient for the routing case.