Applications of duality We have recently covered linear programming and I am comfortable with the weak and strong duality theorems.  However, I don't understand what the applications of duality are that are specific to TCS.  Mike Spivey's blog post gives some interesting examples of some nice properties of duality but I was wondering if there are algorithmic advances, for example,  that come from understanding duality or anything more CS related.
Yup. Everything follows from duality. (I am only half joking). 
A partial list:

Boosting 

The Hard-Core Lemma 

Online Learning 

The ability to actually solve LPs efficiently 

A large fraction of approximation algorithms results.  

Much more

To develop algorithms, you often need a constructive or algorithmic version of the duality theorem (which is essentially equivalent to Von-Neumman's min-max theorem in game theory -- the applications in game theory are also huge). This is essentially what the multiplicative weights algorithm is. See Arora/Hazan/Kale for the best survey on this topic: http://www.satyenkale.com/papers/mw-survey.pdf