Good way to describe co-RE (co-recursively enumerable)?
The class ${\sf coRE}$ contains all languages whose complement is in ${\sf RE}$. Put differently:


A language $L$ is in ${\sf RE}$ if there exists a Turing machine that can check if a requested word $w$ is contained in $L$  for every word $w\in L$. The machine always tells the truth but it may cycle on inputs $w\not\in L$.
A language $L$ is in ${\sf coRE}$ if there exists a Turing machine that can check if a requested word $w$ is not contained in $L$  for every word  $w\not\in L$. The machine always tells the truth but it may cycle on inputs $w\in L$.


An example:
The language $\{ \langle M,w \rangle \mid \text{$M$ cycles on input $w$}\}$ is in ${\sf coRE}$. Just take a Turing machine that simulates $M$ on $w$ and rejects, whenever the simulation stops.