What type of numbers are roots of $x^{2} = -1$ themselves? Are the roots $i, -i$ themselves irrational numbers or complex numbers or left auxiliary so that undefined? 

$i$ and $-i$ are purely imaginary numbers. In symbols, we can write $\Re(x) = 0$.
But the heart of your question is whether these numbers are rational or irrational. Most smart amateurs would say that of course these numbers are rational. Most professional mathematicians would hesitate, and a few would even declare that these numbers are neither rational nor irrational.
To understand that hesitation to extend the definition you need to remember that professional mathematicians don't like to extend definitions unless they can be sure that it's done in a rigorous way and that it does not "break" other definitions.
So how are rational and irrational numbers in $\mathbb{R}$ defined? I suppose first we need to define the integers of $\mathbb{Z}$. But since I don't want to take all day with this, I'm going to assume that you and I agree on the meaning of $1$ and the basic operations of arithmetic. We agree that $1 \in \mathbb{Z}$. Set $m = 1$ and $n = m + 1$, we agree that $n \in \mathbb{Z}$. Reset $m$ to $n$ and do $n = m + 1$ again, we agree that $n \in \mathbb{Z}$. Repeating this procedure forever, we obtain the positive integers. And by doing $n = m - 1$ similarly, we obtain $0$ and the negative integers.
Alright then, the rational numbers. If $x \in \mathbb{R}$ can be expressed as $x = \frac{a}{b}$ such that both $a$ and $b$ are integers from $\mathbb{Z}$, then $x$ is rational, but if not, then it is irrational. (By the way, $0$ is rational, just make sure you don't do $b = 0$).
Moving on to $x \in \mathbb{C}$: we define $i$ so that $i^2 = -1$. But can we express $i$ as $i = \frac{a}{b}$ such that both $a$ and $b$ are integers from $\mathbb{Z}$? We can't, and from that we can come to the "ridiculous" conclusion that $i$ is irrational.
This suggests that we need to extend the definition of rational and irrational to include purely imaginary numbers. Robert Soupe seems to suggest that we consider a number rational if it can be expressed as $\frac{a}{b}$ or $\frac{a}{b}i$. Then $i$ is rational, just as the amateur's intuition indicated.
Extending the definition this way does not seem to create problems, and by problems I mean that it does not suddenly turn some irrational real number into a rational real number.
But it does not address complex numbers with both real and imaginary parts, numbers like $\frac{1}{2} + 14i$. To address such numbers, we could say that a number is rational if it can be expressed as $\frac{a}{b}$ or $\frac{c}{d}i$ or $\frac{a}{b} + \frac{c}{d}i$, with $a$, $b$, $c$, $d$ all drawn from $\mathbb{Z}$. Obviously these are closed under addition, subtraction and multiplication. But are they closed under division? I believe that they are, but I don't want to hold off posting this answer much longer while I slowly verify that, I can come back and edit later.
