How do algebraists intuitively picture normal subgroups and ideals?

Solution 1:

Normal subgroups and ideals are kernels. That is the reason they are interesting, they are precisely the subgroups/rings that are kernels of homomorphisms.

By themselves, ideals and normal subgroups are typically not interesting. They become interesting when considered together with the homomorphisms of which they are the kernels: that is, we are interested in the structure of $G/N$ or $R/I$, and how they, together with the structure of $N$ and $I$, give information about the structure of $G$ and $R$.

Solution 2:

To add to existing answers, ideals in rings are useful beyond being kernels. For example, ideals in a ring $R$ are $R$-modules (abelian groups with an action of $R$) and sometimes have an interesting structure in that sense.

Furthermore, in algebraic geometry we treat the set of all prime ideals in a commutative ring as a topological space with additional structure; this is called the spectrum of the ring. Upon being told it's hard to imagine how this could be a useful topological space; the original motivation is that the set of all polynomials (in any number of dimensions) that vanish on a particular set form a prime ideal. Thus prime ideals could also be seen as points in this sense.