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?

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.
