Hosts send traffic to their default gateway. If that router fails, they cannot reach other networks. First Hop Redundancy Protocols (FHRPs) solve this by letting multiple routers share a virtual IP address.
Clients point to the virtual IP. Behind it, one router is active and handles traffic. If it fails, another takes over the virtual IP. The failover is invisible to clients.
The main FHRPs are HSRP (Cisco), VRRP (open standard), and GLBP (Cisco with load balancing). Each has different features, but all solve the same problem.