The term "radical" I heard in at least four subjects - representations of groups (via group algebra), quadratic forms, Lie algebra and commutative algebra (just now). In the first three subjects mentioned, the basic object are studied modulo radical.
My question is simply, for what purpose the radical of an ideal is considered in commutative algebra/algebraic geometry ?
Here are few quick ideas of where the radical comes into play. I encourage others to contribute too with elaborations and specializations because firstly the radical is a big concept and besides that these are just some things that popped into my head + I'm no expert.
(1) You've probably encountered the concepts of maximal ideas and prime ideals. Quotients by maximal ideals correspond to fields, quotients by prime ideals corresponds to integral domains. Quotients by radical ideals (ideals that are their own radicals) correspond to reduced rings, which are similar to but a bit weaker than integral domains. reduced rings can have zero divisors, but they don't have elements such that $x^n = 0$.
(2) Perhaps you've also encountered Gauss' lemma? In a GCD domain (in which every pair of elements has a greatest common divisor), the product of primitive polynomials is primitive. In the field of Multiplicative Ideal Theory, a major point of interest is in understanding how the coefficients of two polynomials $f,g$ related to the coefficients of their product. The coefficient ideal $A_f$ of a polynomial $f$ is defined to be the ideal generated by it's coefficients of $f$. In Prüfer domains (in which finitely generated ideals are invertible), one has the best behavior: $A_f A_g = A_{fg}$. However, this generalizes to any commutative unital ring in a statement via radicals: with $\sqrt{A}$ denoting the radical of $A$, we always have $\sqrt{A_f}\sqrt{A_g} = \sqrt{A_{gf}}$. See here for details.
(3) Radicals interact well with a lot of other prominent notions in commutative algebra, including prime and maximal ideals, integral closure of ideals, sum/product of ideals. E.g. the radical of the product of ideals is the intersection of their radicals. Radical ideals are integrally closed. Maximal radical ideals are primary. Prime ideals are radical... so you can see how the radical intertwines with the more familiar language of maximal and prime ideals.
(4) I'll go a little bit deeper than (3). An extremely important class of rings are the integrally closed domains, of which GCD domains are an important example, and hence unique factorization domains, bezout domains, prüfer domains, PIDs, and many more galore. I mentioned before that a prime ideal is radical. Conversely, we have that radical ideals are prime in a very important class of rings called valuation domains (in which every pair of elements a,b is such that a|b or b|a, and which admittedly sound weird at first but come up quite a bit in geometric application). I also mentioned before that radical ideals are the intersection of prime ideals containing them. Parallel to this fact is that integrally closed domains are the intersection of all the valuation domains containing them. I'll leave this connection sketchy but I hope its interesting to muse over.
(5) In a word, Nullstellensatz. The concept of the radical is a central part of the formulation of Hilbert's Nullstellensatz, which is the beginning of Algebraic Geometry. The gist is as follows: let $I$ be an ideal in a multivariate polynomial ring over an algebraically closed field. The set of polynomials which vanish at every point that the entire ideal $I$ vanishes is also an ideal, call it $J$. The kicker is that $J = \sqrt{I}$.
The Nullstellensatz and its myriad implications are a lot to unpack, so rather than go further here I'd suggest poking around MSE more or looking at some introductory texts.
The radical is the problematic part of something, in whatever sense one is interested. What exactly the radical is for commutative rings, for ideals, for quadratic forms, for Lie algebras is different, but it always is something modulo which things become better.
