  We prove an effective version of the Pila-Wilkie Theorem for sets definable
using Pfaffian functions, providing effective estimates for the number of
algebraic points of bounded height and degree lying on such sets. We also prove
effective versions of extensions of this result due to Pila and Habegger-Pila .
In order to prove these counting results, we obtain an effective version of
Yomdin-Gromov parameterization for sets defined using restricted Pfaffian
functions. Furthermore, for sets defined in the restricted setting, as well as
for unrestricted sub-Pfaffian sets, our effective estimates depend polynomially
on the degree (one measure of complexity) of the given set. The level of
uniformity present in all the estimates allows us to obtain several diophantine
applications. These include an effective and uniform version of the
Manin-Mumford conjecture for products of elliptic curves with complex
multiplication, and an effective, uniform version of a result due to Habegger
which characterizes the set of special points lying on an algebraic variety
contained in a fibre power of an elliptic surface. We also show that if
Andr\'e-Oort for $Y(2)^g$ can be made effective, then Andr\'e-Oort for a family
of elliptic curves over $Y(2)^g$ can be made effective.
