There are several basic concepts that surround the routing process and the concept of a route:
Routing is the process of moving data from one network to another and is usually performed at the network layer in the OSI model. The routing function is performed by Internet Protocol on networks running TCP/IP.
Any computer with more than one network interface and which has hardware or software that performs routing is called a router. The primary reason for putting a router in a network is to reduce the ammount of broadcast traffic by splitting the network into at least two parts.
Dynamic routing protocols calculate routes by advertising the networks they are connected to and the routes they use to reach other routers connected to more networks. Once all the routers in a network know all routes to all destinations, the routing protocol has said to have 'converged'. If a user were to attempt to reach a destination before the routing protocol converged, the data might never get to its destination. There are two problems that occur when a network is having difficulty reaching the state of convergence: routing loops and black holes.
If a packet reaches a router which does not know how to get to the destination and get dropped, or the packet might get forwarded through a circle of routers in what is called a 'routing loop'. If that happened, the packet would simply run around in circles till it grew old and died. Fortunately, most data is given a time to live. When data is forwarded from one router to the next, one more tick is taken off it's time to live. Eventually that time to live reaches zero and the data is thrown away. This prevents the network from becoming congested.
Black holes are what happens when a route sends data to the wrong place or can't get to the destination. The data is simply dropped and vanishes from the network. Packets go in and they don't come out. Nearly all 'black holes' are unintentional and are caused by problems occurring on the network. However, Internet providers will sometimes intentionally black hole destinations belonging to those who abuse their Internet privledges. SPAMMERS beware!
Why don't networks converge?
When an interface on a router is rendered inoperable for any reason (the interface breaks or burns out, the connector gets bent, the cable breaks etc.), the routes pointing data to that interface are withdrawn from that router's announcements to other routers; thus, convergence can be a difficult thing to achieve on a network with very low reliability.
Advertisements are messages sent between routers that pass on information about how to reach destinations. Some routing protocols advertise the complete list of known routes, some protocols only advertise summarized lists of routes, or changes.
Routing metrics are a scoring system for routes used to indicate how good or bad the route is. Metrics are calculated by routers to prioritize routes from best to worst. Routers use the metrics to select the best possible route or routes to a given destination. Metrics can include hop count (how many stops there are between here and the destination), media type (serial, FDDI, Token Ring, ethernet, SONET etc.), availability (whether the machine is up or down) and several other factors including some set by the Network Administrator. A lower metric generally indicates a better route.