Routing means deciding at each interface in a network where the packet is intended to go. Usually when we talk about routing we mean IP routing. That is the "kind" of routing I will be talking about. IP routing happens at layer three in the OSI model, it has nothing at all to do with the MAC layer. Routing divides broadcast domains. This means that ARP traffic stops at the router. ARP traffic is a layer two function used to discover which MAC host has a specific IP address. In large bridged domains (broadcast domains) ARP traffic can generate substantial network traffic.

Referenced at: http://www.broadbandreports.com/forum/remark,7747719~mode=flat#12003006 by Michael Erskine of Free Antennas.com and posted with permission.

Bridging means repeating at each interface in a network all packets which appear on one side to the other side. A bridge is a repeater. A hub is a "multipoint" repeater. You plug in eight wires on a hub and a packet coming into any wire goes out the other seven. Bridging happens at layer two in the OSI model. That is the MAC layer. It has nothing to do with TCP/IP. As an aside, all this means that a layer two switch is in actual fact a layer two router because, a packet coming in one port on the switch is only repeated on the port where the destination computer is connected. The layer two switch "routes" packets between ports using MAC addresses and the spanning tree protocol. Layer two switching is easily a whole order of magnitude faster than IP routing because IP routing requires a lot more CPU and a lot more software.