Why do we also call the roots of the polynomial that cannot be solved by radicals "algebraic"? Quote from Wiki: "An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients."
But, the root of unsolvable polynomials (by radicals) cannot be obtained by "algebraic" operations:
"addition,subtraction,multiplication, division  and root extraction"

So, if they're not expressible in radicals, in what sense are they 'algebraic'?

 A: A number being the root of a polynomial means you have access to it by the methods of algebra: linear algebra, modules, traces and norms, ideals, and so on. I am not saying that is why the term "algebraic number" was used initially, but the term is very nice today for the reason I just indicated. Non-algebraic numbers are called transcendental since they "transcend" (go beyond) techniques of pure algebra, both historically and today.
You appear to be under the impression that numbers that can be described by explicit root formulas are important, perhaps because a more impressive word "solvable" is used for them. Let me advise you to reconsider this idea. While historically the property of being expressible by radicals was a big deal, and trying to understand this property led to important developments in abstract algebra, after Galois cleared this issue up his ideas (Galois theory) did not atrophy but continued to find many new applications up to the present day. Solvability of a number is not important in modern math while the property of being algebraic is very important: see any account of algebraic number theory.
A: Yes, I, too, "in my youth", was greatly confused about what kind of numbers "algebraic" numbers could be, if they were not expressible in terms of radicals! :)
No, one's high school teachers would most likely not be able to explain the difference...
So, yes, the obvious question "but, wait, if they're not expressible in radicals, in what sense are they 'algebraic'?" deserves an answer... which is that "it turns out" that there are roots of polynomial equations not expressible in terms of radicals. Who knew? But they're still characterized by purely algebraic means (just not quite expressions in terms of radicals).
In the 19th century, and into the 20th, there was considerable interest in determining what "new" operations, beyond root-taking, would be necessary to express solutions to polynomial equations. The "Tschirnhausen transformations" showed that there was some meaningful reduction. In the late 19th, Klein (and others) showed that modular forms could express solutions of quintics. In the 20th, people realized that, similarly, Siegel automorphic forms (connected to the Jacobian variety of an algebraic curve attached to a polynomial) could express the solution of an arbitrary polynomial. (See Mumford's three volumes "On Theta"... or similar title.)
(And, in the 20th, we did also see that elliptic functions, and so on, can express Hilbert class fields of certain number fields. Work of Shimura, et al. And then there are Stark's conjectures about generation of fields...)
