Open Most Complicated Routing Protocol First

OSPF! Open Shortest Path First.  It's an interior gateway routing protocol - you'd use OSPF within a given AS, for instance.  It has a whole bunch of things that can be tweaked.  Most people (in my experience) use it as their interior routing protocol.

OSPF is based around the idea of a link state diagram -- that is, a diagram of the network connections between all the routers in a given OSPF instance.  When making routing decisions, a given router will figure out what the best ("shortest") next-hop is for a given destination.  If left untweaked, the route metric will be based on the number of hops and the bandwidth of each link (lower-bandwidth links are less preferred).  You can adjust that and manually weight routes to encourage traffic to go over a path that it would otherwise not prefer.

An OSPF-enabled router uses broadcast packets to announce itself and to find out about other OSPF routers in the same collision domain.  If compatible, the routers will establish a neighbor relationship, and exchange route information.  Among neighbors, there's a designated router and a designated backup router.  The designated router is a given router interface that is in charge of sending out link state announcements and letting the other router interfaces know what's up with the links.  It's a specific interface, not a specific individual router -- a router that is a DR on interface 1 might not be the DR for a separate OSPF instance on interface 2.