Nothing on the web; What is a Ruffini Radical Surprisingly, it's not clearly defined online. The first thing that comes up is Abel-Ruffini theorem, which only refers to "radicals" and not RUFFINI radicals.
Ian Stewart's book has it appear out of thin air as if it's prior knowledge and common to all readers. Unfortunately, I have no memory of learning this.
Given its fancy name, I am sure it's different from radicals in general(otherwise, it's very un-math-like to do something meaningless like giving something a fancy name just for the sake of it).
What is the concrete definition of it?
 A: Sigh. He is defining a phrase "soluble by Ruffini radicals." That is what is in italics. It is not clear that he will have any use for the shorter phrase "Ruffini Radicals." 
He begins with 

The next definition is not standard, but its name is justified because
  it reflects the assumptions made by Ruffini in his attempted proof
  that the quintic is insoluble.

and then provides the definition

DEFINITION 8.8. The general polynomial equation $F(t)=0$ is
  soluble by Ruffini radicals if there exists a finite tower of subfields
  $$\mathbb{C}(s_{1},\ldots,s_{n})=K_{0}\subseteq K_{1}\subseteq\cdots\subseteq K_{r}=\mathbb{C}(t_{1},\ldots,t_{n})\tag{8.6}$$
  such that for $j=1,\ldots,r$,
  $$K_{j}=K_{j-1}(\alpha_{j})
    \qquad\text{and}\qquad \alpha_{j}^{n_{j}}\in K_{j}
    \qquad\text{for}\qquad n_{j}\geq2,\ n_{j}\in\mathbb{N}$$
Source: Stewart, N. I., Galois Theory. Fourth edition. CRC Press (2015).

Note that "soluble by Ruffini radicals" as a whole is italic.
A: A Ruffini radical in this context means that it isn't allowed to be of a form so that it "brings in" any radicals (n-th roots) from anywhere else than the field generated by the roots (and the polynomial coefficients, which is a subfield). The non-Ruffini radical fields that Stewart implicitly talks about don't have that limitation but can be pulled from anywhere. They don't have the limitation $K_j \subseteq \mathbb{C}(t_1,...,t_n)$.
Stewart says on page 118 in the 4th edition (about Ruffini radicals): "The aim of this definition is to exclude possibilities like $\sqrt{-121}$ in Cardano's solution of the quartic equation $t^4-15 t-4=0$, which does not lie in the field generated by the roots, but is used to express them by radicals."
A section or two forward in the book, after the mentioning of Ruffini radicals, Stewart has included a theorem and proof of Abel (difficult) that shows that as a matter of fact bringing in roots from "anywhere else" doesn't bring any more power to the game. So Ruffini radicals turns out to be all you need. This blog talks about this: https://jmanton.wordpress.com/2016/08/11/a-snippet-of-galois-theory/
Doing Abel's proof using Galois theory (which didn't exist when Abel did his proof) is much easier and is covered by, for example, Wikipedia: https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
Stewart shows how to do it "the hard way" without Galois theory, the way Abel did it. This is what makes it necessary to introduce "Ruffini radicals".
(Note that some editions of Stewart's book seem to have a typo. In the definition of Ruffini solubility it should say $K_{j+1}=K_j(α_j)$)
