How should I read and interpret $A = \{\,n^2 + 2 \mid n \in \mathbb{Z} \text{ is an odd integer}\,\}$ As my question states, I need to interpret: $A = \{\,n^2 + 2 \mid n  \in \mathbb{Z} \text{ is an odd integer}\,\}$
Does this mean that any odd integer $n$ will work, or does this mean that the output of $n^2 + 2$ must be an odd integer?
For example, would these numbers be in the set: $\{1,3,5,7,\ldots\}$ or would $5$ not be in the set because $2^2 + 2 = 6$? 

The condition on the right of the "$|$" is not about what is on the left of it. So for each $n\in \Bbb Z$ that is an odd integer (that is, for $n=\ldots,-5,-3,-1,1,3,5,\ldots$) we form the expression $n^2+2$ (that is, $\ldots, 27,11,3,3,11,27,\ldots$) and collect the results in the set $A$. In orther words, $$A=\{3,11,27,51,83,\ldots\}.$$
(Incidentally, in the given situation $n^2+2$ is odd if and only if $n$ is odd; the distinction to be made might be clearer if we considered $\{\,n^2+1\mid n\in\Bbb Z\text{ is an odd integer}\,\}$)

Some remarks on this notation:
Admittedly, in a strict sense when introducing set theory axiomatically, one usually defines the followng two set-builder notations (used in the Axiom Schema of Comprehension and the Axiom Schema of Replacement, repsectively):
$$\tag1 \{\,x\in S\mid \Phi(x)\,\}$$and$$\tag2\{\,f(x)\mid x\in S\,\}$$
(where $S$ is a set, $\Phi$ is a predicate, and  $f$ is a function).
These are determined  by 
$$ a\in \{\,x\in S\mid \Phi(x)\,\}\iff a\in S\land \Phi(a)$$
and 
$$ a\in\{\,f(x)\mid x\in S\,\}\iff \exists x\in S\colon f(x)=a.$$
What you have is a mix of these, i.e., is of the form $$\tag3\{\,f(x)\mid x\in S\land \Phi(x)\,\}$$ and thereby possibly confusing. To follow the notational convention strictly, we need $x\in\text{(some set)}$ on the right if we want to use a function on the left, so should write something like $\{\,f(x)\mid x\in\{\,y\in S\mid \Phi(y)\,\}\,\}$, but in my opinion that would be less legible (and thereby more confusing):
$$ A=\bigl\{\,n^2+2\bigm|n\in\{\,k\in\Bbb Z\mid k\text{ odd}\,\}\,\bigr\}$$
At the same time this way to rewrite $(3)$ in terms of $(1)$ and $(2)$ shows that introducing $(3)$ as a shorthand(?) can be justified. 
Also, since it is clear in your example that the function values are themselves in an already well-known set (namely $\Bbb Z$), we could get by with a notation that uses only comprehension, not replacement:
$$ A=\{\,m\in\Bbb Z\mid\exists n\in\Bbb Z\colon(n\text{ odd}\land m=n^2+2)\,\}.$$
Again, this probably does not lead to more enlightenment about $A$ than the given notation, so is not necessarily really a notational improvement.
Since oddness can be easily expressed ($n=2k+1$ with $k\in\Bbb Z$), I'd prefer
$$ A=\{\,(2k+1)^2+1\mid k\in\Bbb Z\,\}.$$
