How is N defined? I understand that $\mathbb{Z}$ and $\mathbb{Q}$ are defined as ...
$$\mathbb{Z} = \mathbb{N} \cup \{ -n \mid n \in \mathbb{N} \}$$
$$\mathbb{Q} = \left\{ \frac{a}{b} \mid a,b \in \mathbb{Z} \right\}$$
... but how is $\mathbb{N}$ defined?
(and how is the order on these sets defined?)

You can make a construction of natural numbers based on set theory by defining $0:=\{\,\}=\varnothing$ and the successor of $n$ $($denoted by $S(n))$ as $S(n)=n\cup\{n\}$. And so for instance :


*

*$1:=S(0)=\varnothing\cup\{\varnothing\}=\{\varnothing\}$.

*$2:=S(1)=\{\varnothing\}\cup\{\{\varnothing\}\}=\{\{\varnothing\},\{\{\varnothing\}\}\}$.


Now, for what concerns the ordering: Let $a,b\in\mathbf{N}$, then $a\gt b$ if and only if $b\in a$. 
