What counts as a standard model of arithmetic? In my research so far, I've found that the canonical standard model of arithmetic is $\mathbb{N}$ under the addition and multiplication operations. However, I've been unable to find much on any other standard model of arithmetic.
Is $\mathbb{N}$ the only standard model of arithmetic, or are there others? I intuitively imagine that $\mathbb{Q}$ under addition and $\mathbb{R}$ under addition and multiplication are standard models, but I've yet to find any support for this. Is a model either standard or nonstandard, or is there an in-between where stuff like $\mathbb{Q}$ and $\mathbb{R}$ would go? If there's more than one, then what precisely defines a standard model of arithmetic?

The ordinary natural numbers, under the usual addition and multiplication, are a standard model of arithmetic. Any model not isomorphic to these is called non-standard. The usual convention in logic is that by $\mathbb{N}$ we mean the numbers $0,1,2,\dots$. 
We have left the term arithmetic undefined. One usually has a particular theory in mind, such as first order Peano arithmetic. Or perhaps the theory whose axioms are all sentences of the usual language of arithmetic that are true in $\mathbb{N}$. That is a theory incomparably stronger than first order Peano arithmetic. 
The term arithmetic may be confusing here. The word has many meanings. In particular, it is an old-fashioned term for number theory. That is what arithmetic means when we refer to non-standard models of arithmetic.
Remark: The structures $\mathbb{Q}$ and $\mathbb{R}$, under the usual operations, are not models of arithmetic. To see this, let $\varphi$ be the sentence $\forall x(\lnot(x=0)\longrightarrow \exists y(xy=1))$. Then $\varphi$ is false in $\mathbb{N}$, but true in $\mathbb{Q}$ and $\mathbb{R}$. There are many other similar sentences. 
