An area is referred to as a stub area when there is only a single exit point from that area and traffic can only enter and leave the area through a single Area Border Router (ABR). A stub area will still have routes to networks within other areas of the OSPF Autonomous System (AS), but will only have a single default route for routes external to the OSPF AS.
A totally stubby area takes the above one stage further by only injecting a single default route for all routes external to the stub area.
One reason to use stub/totally stub areas is to reduce the resource requirements i.e., memory, CPU etc., for routers within the stub area.
Take a look at http://www.cisco.com/warp/public/104/1.html#t31 for further info on stub and totally stubby areas.
There are certain restrictions when using stub areas, one of which is that an Autonomous System Border Router (ASBR) can not exist within a stub area. If then a router within the stub area is learning routes from some other routing protocol i.e., RIP, it will have no way of injecting those routes into the OSPF AS. To overcome this limitation the concept of the NSSA was introduced.
Take a look at http://www.cisco.com/warp/public/104/nssa.html for further information on the NSSA
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