ios_base is a class at the top of the class hierarchy in the IOStreams library.
The basic_ios<> class template derives from ios_base, and therefore constitutes the 2nd rung in this hierarchy.
Now, streams can be configured to throw exceptions which are of type ios_base::failure, which is a class.
Since ios_base::failure is a class and ios_base is also a class, I must conclude that ios_base::failure is a class defined _within_ the ios_base class. (A class defined within another class)
std::ios_base::failure may be defined either as a member class of std::ios_base or as a synonym (typedef) for another class with equivalent functionality