The definition as it appears in Yosida's book:
A locally convex space $X$ is called bornologic if it satisfies the condition: If a balanced convex set $M$ of $X$ absorbs every bounded set of $X$, then $M$ is a neighborhood of $0$ of $X$.
An alternative definition is provided by (also from Yosida):
A locally convex space $X$ is bornologic iff every seminorm on $X$, which is bounded on every bounded set, is continuous.
I'm having trouble understanding how to think of either of these definitions.
I see from wikipedia that bornologic spaces are defined like they are to be able to ask questions about boundedness of sets or functions. But isn't there already a satisfactory definition of 'bounded set' only in the context of topological vector spaces (namely that a set is bounded if it is absorbed by any neighborhood of the origin)? What extra do bornologic spaces provide?
Also, what are some typical examples of spaces which aren't bornologic? I know that every Fréchet space is bornologic, so it seems like in practice most spaces should be bornologic. Is this true?