Need help understanding what co-recursively enumerable means Lets say I have a set: $ L = \{\langle G \rangle | L(G) = \Sigma^{\star}\}$ and the question asks if it is co-RE. I know that if something is co-RE, it halts on every input not in L but may or may not halt on inputs in L. 

So in this case would something "not in L" be nothing?
If you by co-CE mean co-Computably Enumerable, then they are the same.

The field called Recursion Theory is also known as Computation Theory.

You find the same analogy in the classes.  Recursive is a different way of calling Computable.  A recursively enumerable set is a different way of calling computably enumerable set.