What does it mean to gauge a group? I'm starting to learn about gauge theories and Goldstone bosons.  What does it mean for a group to be gauged?
 A: Begin with theory of fields $\phi$ on a manifold $M$ with target space $V$ (which we take to be a vector space in this discussion)
$$
  \phi:M\to V
$$ 
that exhibits a global symmetry under some group $G$.  This means that there is a representation $\rho$ of the group $G$ acting on the target space $V$ under whose action the theory is invariant, in other words the transformation
$$
  \phi(x) \to \rho(g)\phi(x)
$$
is an invariance of the theory.  By gauging the group, we mean that we allow for transformations of the fields under this representation for which the group element $g$ depends on the point $x$ on the manifold where we are applying the transformation;
$$
  \phi(x)\to\rho(g(x))\phi(x)
$$
Such a transformation is called local.  We now look to see how we can modify the theory (the standard way is to include an auxiliary field called a gauge field along with a gauge covariant derivative) such that the new theory exhibits invariance under these local transformations.  The result is a gauge theory.
A: It means that instead of global transformations
$ \Psi(x,t) \rightarrow U \Psi(x,t) $
for some group element $U$, you allow local transformations
$ \Psi(x,t) \rightarrow U(x,t) \Psi(x,t) $
where $U(x,t)$ is a space-time dependent group transformation. Any textbook on quantum field theory will go into detail about the implications of this. :) A free pre-publication version of Srednicki's (pretty good) book is available here if you don't already have access to one.
