Noether's Theorem for Classical Fields

We start with a classical theory of fields $\phi$ with (local?) action $S$. Noether's theorem for classical fields says that every symmetry of the action yields a conserved quantity. In order to understand and prove the theorem then, we need to define the terms "symmetry" and "conserved quantity."

When we generally say that a certain object is symmetric, we almost always mean that the object in question remains the same after we perform a transformation on it. For example, we say that the two-sphere $S^2$ is rotationally symmetric because when we apply any rotation whatsoever to it, it does not change. Whenever something doesn't change after a transformation is applied to it, we say that it is invariant under that transformation. So in general we say that a symmetry of something is a transformation under which it is invariant.