What does it mean for a set $A$ to be computably enumerable in another set $B$? In my introductory computability class I keep seeing the phrase set $A$ computably enumerable IN set $B$? 
I don't want a definition of computably enumerable, I know what that is, and there are a lot of answers for that. I purely want to know why one set is somehow being INSIDE another one means. 
An example of the phrasing being used when talking about Turing degree:

(iii) A degree a is computably enumerable in degree b (a is c.e. in b) if there exists some set $A \in \mathbf a$ that is c.e. in some set $B \in \mathbf b$.

 A: "In" here doesn't mean "inside of" - it's more in the spirit of "in a parameter for." Really, the phrase we should use in place of "in" is "relative to" - if you're familiar with Turing reducibility, saying "$A$ is c.e. in $B$" is analogous to saying "$A$ is computable relative to $B$," or "$A\le_TB$."
If you're not familiar with Turing reducibility, we're talking about oracle machines. Recall that $A$ is c.e. if $A$ is the set of inputs on which some Turing machine halts; well, $A$ is c.e. in $B$ if $A$ is the set of inputs on which some oracle Turing machine halts when given $B$ as an oracle. More snappily:

"$A$ is c.e. in $B$" means that using $B$ as an oracle we can computably enumerate $A$.

For example, it's easy to see that $A$ is always c.e. in $\overline{A}$; by contrast, the complement of a c.e. set is c.e. iff it is computable. More interestingly, we can relativize the Halting Problem and this gives us - for any $B$ - a "maximally complicated" set $A$ which is c.e. in $B$, analogously to how the Halting Problem itself is the "maximally complicated" c.e. set.
