Why are amidines so much more basic than amides?
Here is a diagram comparing what happens when we protonate an amide and an amidine.



Let's compare the protonated products in the two reactions.  In the case of the amidine we can draw two important (the fact that they are equivalent resonance structures is why they both are important) resonance structures to describe the protonated product.  In the case of the protonated amide the resonance structure with the double bond to oxygen places a positive charge on oxygen.  Therefor this resonance structure is a less significant contributor than the corresponding structure for the protonated amidine where the positive charge resides on the less electronegative nitrogen.  Since we can draw more significant resonance structures for the amidine case, the protonated amidine is more stable than the protonated amide.  This difference in protonated product stabilities explains why amidines are more basic than amides.