I'm trying to learn the theory of algebraic groups using "modern" algebraic geometry, going past the traditional, very standard books by Borel, Springer, and Humphreys, all of which try to avoid the use of group schemes. One of the motivations for using schemes is that in the classical setting, "nilpotents are not allowed," a sentence I don't understand. For example, on page 18 in Milne's Algebraic Groups (which is still in draft form), he writes, "Since we allow nilpotents in the structure sheaf, the points of an algebraic group with coordinates in a field, even algebraically closed, do not convey much information about the group. Thus, it is natural to consider its points in a $k$-algebra. Once we do that, the points capture all information about the algebraic group."
This is clearly not meant to be totally precise. But I still wonder what he means. I'm thinking of the curves $x^2=0$ and $x=0$ as one example which is sometimes used to motivate the definition of scheme (the behavior at 0 is different in those two examples).
Here are my questions, presumably related: By "nilpotents in the structure sheaf," does he mean nilpotents in the coordinate ring? What does "allowing nilpotents" mean, and what is wrong with not allowing nilpotents? Why doesn't allowing nilpotents force the points to "convey much information" about the group? What information does thinking of algebraic groups as functors capture?
I should add that I'm mostly used to thinking of algebraic groups as functors. For example, $SL_n$ is thought of as a functor that takes a ring $R$ to the matrix group $SL_n(R)$.
(I can post this as more than one question if it would be better, but I don't know enough about the answer I'm looking for to know whether or not that is a good idea.)