What is a covariant derivative in gauge theory? I've been studying electroweak theory and you need to keep the Lagrangian covariant by introducing covariant derivatives. What is a covariant derivative? And what does it mean to keep the Lagrangian covariant?
Also, in electroweak symmetry breaking, the gauge bosons attain their masses via the action of a 'covariant derivative' on the Higgs field. What does this mean in physical terms?
 A: Further to John's suggestion in his comment, also look up the Wiki article for gauge covariant derivative. 
A covariant derivative is an extension of the "rise over run" co-ordinate derivative so that its "output" is something that transforms like a tensor (and so is one) so that it is independent of co-ordinates. Closely linked is the idea of parallel transport: tangent spaces to manifolds are generally not comparable (unless the manifold is Euclidean) - witness for example the the different tangent planes on a sphere: you can't sensibly subtract vectors in one from those in another because they're not the same vector space: they are tilted to one another. Now, of course, you can subtract them by thinking of the sphere embedded in the higher dimensional Euclidean 3 space, but in general we don't want to do such embedding: the manifold we're dealing with may be a configuration space for some physical system which has no really meaningful embedding in a higher dimension Euclidean space. So we need to define a parallel transport map that maps a tangent vector in one tangent space to its corresponding, "parallel transported" vector in another. We define this latter as the "no change" version of the former. We then define the "difference" of vector field changing between the two tangent spaces in question to be the difference between its value at the latter tangent space less of the parallel transported version of its value in the former tangent space.
See also my description of parallel transport in my answer here.
It turns out that one of the smoothest ways to do this in a way that does not depend on coordinates used is to do it abstractly, defining a covariant derivative by certain general properties like (see the "Formal Definition" in the Covariant Derivative Wiki):


*

*A covariant derivative of a scalar simply the wonted coordinate derivative;

*A covariant derivative of a vector must be a linear operator on the tangent space;

*A covariant derivative of vector and a scalar fulfills the Leibniz product rule;

*Covariant derivatives of tensor products also fulfill a natural generalisation of the Leibniz product rule.


Wulf Rossmann in his Lecture notes on differential geometry deals with these ideas very clearly in chapter 3.
A gauge covariant derivative derives its notion of parallel transport in such a way that the derivative is invariant under gauge transformations. This is actually more general than the above: it reduces to the above when we consider the gauge invariance to be invariance under diffeomorphism. In other words, we need it to transform as a tensor.
