what is a quartic non-biquadratic CM field? That's my question: what is what people call a quartic non-biquadratic CM field? 
Is that the easiest example of a CM number field whose Galois closure is non-abelian? 
 A: It sounds like you might be looking for something like $\mathbb{Q}(\sqrt{\sqrt{2}-3})$. It is degree $4$ over $\mathbb{Q}$. It is totally imaginary and is a quadratic extension of a totally real field (namely $\mathbb{Q}(\sqrt{2})$). The Galois closure is dihedral of order $8$. 
In general, a quartic CM field will have to be of the form $\mathbb{Q}(\sqrt{a+b \sqrt{D}})$ with $D>0$ and with $a \pm b \sqrt D < 0$ for both choices of sign. (In general, a quadratic extension of a quadratic field is of the form $\mathbb{Q}(\sqrt{a+b \sqrt{D}})$, and the sign conditions correspond to the CM condition.
This will usually have Galois group $D_{2 \times 4}$, but it will sometimes specialize to have Galois group $\mathbb{Z}/2 \times \mathbb{Z}/2$ or $\mathbb{Z}/4$. So presumably, a "quartic non-biquadratic CM-fields" means a quartic CM-field where the group doesn't become $\mathbb{Z}/2 \times \mathbb{Z}/2$.
I haven't heard this concept singled out for attention before, so I can't give any context as to why this is an important class of fields to consider.
