A bridge is a relatively simple device. Its only purpose is to split the physical network being shared by many computers into smaller segments. A bridge generally has only two ports; bridges with more than two ports are usually called switches.
Since Ethernet is the most commonly used physical network, we'll use it to illustrate our point. On an Ethernet network, all the connected computers share the same piece of "wire" (it's not physically the same piece, but it is electrically). When two computers attempt to talk at the same time, they drown each other out and create what's called a collision. The more computers you have on one Ethernet, the bigger chance you'll have a collision.
Bridges split an Ethernet into multiple collision-domains. All the data on one side of the bridge stays there, unless it's destined for a computer on the other side of the bridge, lessening the overall load on each segment.
Bridges don't care what protocol is being used on the network (TCP/IP, IPX, AppleTalk, etc.) since they operate at what's called the data-link level. This is both a benefit and a curse; since they work at such a simple level, bridges are able to operate at blindingly fast speeds, but since they will indiscriminately forward data, one has little control over their operation. This is where routers come in.
Routers work at the network layer - they actually understand the protocols being used to carry the data over the network. And since they understand the protocols, they can use rules to decide what to do with a specific piece of data. Because of this, routers are useful in linking networks that are used for different purposes or by different organizations. One can apply rules or filters to let certain data in, but keep other data out. Or to route data serving one purpose over a certain set of network connections, while routing other data over other connections.
This convenience comes at a price. The more detail a router must acquire about a specific piece of data before forwarding it on, the longer that piece of data is delayed before being sent on to its destination. Also, the greater configurability of routers requires faster, more expensive hardware.
All said, in most of today's larger networks you'll find a mix of switches (basically a multiport bridge) and routers. Both have their strengths and weaknesses - but together they make a potent match.
