C-symmetry means the symmetry of physical laws over a charge-inversion transformation. Weak interactions violate C-symmetry maximally. But see left-right models.
The laws of electromagnetism (as well as weak interactions, both classical and quantum) are invariant under this transformation: if each charge q were to be replaced with an equivalent opposite charge -q in all of the equations of electromagnetism, the laws would preserve the same form. However, our universe is not invariant under charge conjugation (it's possible for a solution to be less symmetric than the laws describing it), owing to the fact that (at least the nearby part of) our physical universe is made up of matter and not anti-matter. See baryogenesis. Though, in principle, the choice of 'positiveness' or 'negativeness' is only a convention, in reality asymmetry of charge is an observed fact.
It was believed for some time that this could be combined with the parity-inversion transformation (see P-symmetry) to preserve a so-called CP-symmetry. However, violations of even this symmetry have now been identified. See CKM matrix. The Wightman axioms and local quantum physics predict that CPT-symmetry, which also factors in time-reversal is, indeed, a symmetry.
In quantum field theory, C corresponds to a charge conjugation of the fields.
There is really a lot of ambiguity and arbitrariness in the definition of charge conjugation. To give an example, take two real scalar fields, φ and χ. Formulated as it is, both fields have even C-parity. Now reformulate things so that 
. Now, φ has an even C-parity wheareas χ has an odd C-parity. But let's redefine 
. Now it's the other way around. Similarly, a complex Weyl spinor can be reexpressed as a real Majorana spinor and vice versa. It's really this arbitrariness which allows physicists to define C the way it is in left-right models.