Static Routing
Static routing is not really a routing protocol. Static routing is simply the process of manually entering routes into a device's routing table via a configuration file that is loaded when the routing device starts up. As an alternative, these routes can be entered by a network administrator who configures the routes manually. Since these manually configured routes don't change after they are configured (unless a human changes them) they are called 'static' routes.
Static routing is the simplest form of routing, but it is a manual process.
Use static routing when you have very few devices to configure (<5) and when you know the routes will probably never change.
Static routing also does not handle failures in external networks well because any route that is configured manually must be updated or reconfigured manually to fix or repair any lost connectivity.
Dynamic Routing
Dynamic routing protocols are supported by software applications running on the routing device (the router) which dynamically learn network destinations and how to get to them and also advertise those destinations to other routers. This advertisement function allows all the routers to learn about all the destination networks that exist and how to to those networks.
A router using dynamic routing will 'learn' the routes to all networks that are directly connected to the device. Next, the router will learn routes from other routers that run the same routing protocol (RIP, RIP2, EIGRP, OSPF, IS-IS, BGP etc). Each router will then sort through it's list of routes and select one or more 'best' routes for each network destination the router knows or has learned.
Dynamic routing protocols will then distribute this 'best route' information to other routers running the same routing protocol, thereby extending the information on what networks exist and can be reached. This gives dynamic routing protocols the ability to adapt to logical network topology changes, equipment failures or network outages 'on the fly'.