What does it mean to renormalize an effective field theory? This is in reference to slide 19 of this talk
"As always in Effective Field Theory, the theory becomes predictive when there are more observables than parameters" 


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*Can one explain what this exactly means? May be if you can refer me to examples or literature where this is explained?

*In this case it seems that this translates into having 2 fields ($\delta$ and $v$) and hence 3 2-point functions to regularize but one seems to have only two counterterms. Then how does this make sense? 
Isn't the quoted line basically mean that there is a meaningful interpretation of the scenario with having more correlations to regularize than counter-terms? What is the meaning? 
 A: Theories like QED, where one has a finite number of relevant operators are very rare. Many important predictions are performed by means of effective theories (Please see the following reviews by: Aneesh V. Manohar and Scherer and Schindler). It is sometimes said that that since these theories are nonrenormalizable,  then any loop correction beyond the tree level is not useful because we can actually have an infinite numbers of free parameters that can  fit the theory to any data that we have. 
However, Consider for example chiral perturbation theory which describes the low energy degrees of freedom of QCD (with 3 flavors):
$$ \mathcal{L} = \frac{1}{4} f_{\pi}^2\partial_{\mu}U^{\dagger}\partial_{\mu}U + ...$$
($U \in SU(3)$ is generated by the meson fields). Here, one loop corrections give rise to 8 counterterms whose coefficients can be estimated from various processes. There is evidence for improvement in the precision with respect to the tree level computation.
The improved predictions can be explained as follows: Even though the theory is not renormalizable in the usual sense, but if we restrict the
Lagrangian to terms with less than a given number of derivatives, and a given number of loops, then, there is a finite number of counterterms. The example mentioned above corresponds to terms with up to 4 derivatives. The same terms of order 4 also serve as counterterms needed to renormalize the one loop contribution of the terms with up to 2 derivatives. Thus, if we limit ourselves low energy processes, we need only a finite number of counterterms. In other words up to a given energy scale, we have control on the counterterms
In the example of chiral perturbation theory, we know that it is a low energy effective theory, thus we know that we should stop at some level of the number of derivatives (or momenta). 
This procedure is known as approximate renormalizability, in contrast
to the "exact" renormalizability present in QED, where any energy scale can be reached.  Actually, this exact renormalizability is not of much practical use, since QED itself is not valid to very high energies (other interactions become important).
Thus given that we want to work up to some energy scale, we can treat the effective field theory as a renormalizable theory and perform  loop expansions which generate counterterms with no higher scale. 
The question is how can we know where to stop. The answer lies in our knowledge of the degrees of freedom outside the theory. For example, the Fermi theory of weak interactions (which includes an effective four fermion term) gives good predictions for beta decays up to energies of the order of magnitude of the mass of the $W$-boson which is integrated out in the Fermi theory.
This "relaxation" of the renormalizability requirements does not mean
that we have an infinite number of effective theories at our disposal.
Extra structures are needed to create the property of approximate renormalizability. For example, chiral perturbation theory stems from the origin of pions as the Goldstone bosons of the chiral symmetry breaking. 
A major factor which can spoil the approximate renormalizability are anomalies. If we try to gauge an anomalous symmetry, then we loose control on the number of derivatives in the counterterms. Moreover, if we gauge only anomaly free subgroups, then the counterterms will be gauge invariant up to total derivatives in the Lagrangian, and we have Ward identities to each scale. 
