The impossibility of solving the general polynomial of degree $\ge 5$ by radicals is surely one of the most celebrated results in algebra. This result is known as the Abel-Ruffini theorem, although it's usually asserted that Ruffini's proof was incomplete.
Still, Paolo Ruffini's contributions to algebra seem to be neither widely known nor well understood. This is perhaps not so surprising since Ruffini's first attempted proof spanned 516 pages and the mathematical argument was difficult to follow.
A modern discussion of Ruffini's proof is found in Ayoub's article 'Paolo Ruffini's contributions to the quintic'. Accoring to Ayoub, Ruffini's argument was not a priori flawed, but it relies on several unproved non-trivial claims. More precisely, Ruffini fails to prove that the splitting field is one of the fields in the tower of radicals which corresponds to a solution expressed in radicals.
The revolutionary proofs of Abel and Galois following Ruffini paved way for group theory and Galois theory. Still, one could wonder what Ruffini actually proved in those 516 pages:
Did any significant new mathematical concepts, ideas or theorems arise from Ruffini's work on the quintic?
Cauchy seems to be one of the few mathematicians who found inspiration from Ruffini's work. In a letter dated 1821, he writes:
"... your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgement, proves completely the insolvability of the general equation of degree $>4$. [...] In another memoir which I read last year to the Academy of Sciences, I cited your work and reminded the audience that your proofs establish the impossibility of solving equations algebraically ..."
In fact, according to Pesic, Cauchy's influential 1815 paper on permutations is clearly based on work of Ruffini. So are there other examples of his influence?