Question:   Would you consider the conjugate base of carbonic acid (HCO₃^-1) to be a conjugate base or a weak acid?  Seems to me like there is a contradiction since it can donate an extra proton (although I believe it is more difficult to remove then the first proton) and the pH is about 6.25.

Answer:

To answer your question, let's take a look at the carbonic acid example (other polyprotic acids such as phosphoric acid do pretty much the same thing):

Your question is this: is the bicarbonate ion the conjugate base of carbonic acid or a weak acid in itself?

The answer:  It's both!  Because it's formed when carbonic acid loses a hydronium ion, it is, by definition, a conjugate base of carbonic acid.  Because it has a hydrogen atom on it, it has, by definition, some acidity.  The same thing goes with the H₂PO₄^-1 and HPO₄^2- ions that are formed when phosphoric acid loses a hydrogen atom.

Incidentially, the idea of acidity isn't limited to things that have a pH of less than 7.  Anything that has a hydrogen atom on it can potentially give it off, behaving as an acid.  Ammonia (usually thought of as a base) can behave as an acid to form NH₂^-, though it doesn't happen much.  As you might imagine, it's a very, very weak acid, and a much better base than an acid.