What is a Turing Machine in class coNP
The easiest way (for me) to understand co-NP is as the class of problems where certificates for "No" answers can be quickly verified (i.e. certificates of non-membership).

So if we look at NP as a class of nondeterministic Turing Machines which quickly (in polynomial time) determine that their input satisfies some property $\Pi$, co-NP is the class of nondeterministic Turing Machines that quickly determine that their input does not satisfy $\Pi$ (or equivalently satisfies $\neg\Pi$).