A router may receive multiple paths to a destination. Only the best paths will be entered into a routing table; this is decided by some algorithm defined by the specific Dynamic routing protocols in use.
In general, each possible path will be assigned some value to indicate the path’s preference. The lowest value assigned will then be the preferred path (think of the value as the cost of getting to a destination: the lower the cost, the better).
Note
Each router will take information its neighbours or from the topology itself and make its own decisions.
Whenever a best path to a destination is lost (i.e. a link goes down), routers will automatically remove the path from the Routing Table and replace it with the next best route.