Let me give a unifying example from the "structural," as opposed to algebraic, side: **varieties of proof by induction**.

The natural numbers are well-ordered: every (nonempty) subset has a least element. In other words, **proof by induction works in $\mathbb{N}$.**

When we go past $\mathbb{N}$, we still get versions of induction, but they become progressively more complicated:

$\mathbb{Z}$ still supports a version of proof by induction, but we have to work "in both directions:" "induction" in $\mathbb{Z}$ looks like

If $P$ is true of $0$, and $P(n)\implies P(n+1)$, and $P(n)\implies P(n-1)$, then $P$ is true of all integers.

$\mathbb{Q}$ *also* supports a kind of induction, only now it's even more complicated. The idea is to look at how rational numbers are "built up." The following is true:

Suppose $(i)$ $P$ is true of $0$, $(ii)$ $P(n)\implies P(n+1)$, $(iii)$ $P(n)\implies P(n-1)$, and $(iv)$ $P(n), P(m)\implies P({n\over m})$ (for $m\not=0$). Then $P$ is true of all rational numbers.

What about $\mathbb{R}$? Well, now we're in a pickle: **$\mathbb{R}$ is uncountable**, so there's not going to be a similar way to "build" the real numbers starting from $0$ and just using a few basic operations *(that will only ever get countably many things - as long as "few" means "at most countably many")*. However, something really cool happens:

Moving from $\mathbb{Q}$ to $\mathbb{R}$, we actually **gained** an important property: **the least upper bound property**, that every (nonempty) set of real numbers has a least upper bound. This is really the fundamental property of the real numbers, and is used for basically everything. One really interesting consequence is that the real numbers allow the following "real induction:"

Suppose $P$ is a property such that for all $X\subseteq [0, 1]$, if $P$ holds for every element of $X$ then $P$ holds for $\sup(X)$ *(or rather, the least upper bound *in $[0, 1]$* of $X$ - the difference is that $\sup(\emptyset)=0$, giving our base case, rather than undefined)*. Then $P$ holds for every element of $[0, 1]$.

Note that this principle doesn't even make sense for $\mathbb{Q}$, since sets of rationals need not have rational least upper bounds. *(I've stated real induction for the interval $[0, 1]$ above, since the most famous application of topological induction is a snappy proof of the Heine-Borel theorem, but you can easily rephrase it for the nonnegative reals, or for all reals (a la the $\mathbb{Z}$-example).)* Proving real induction from the least upper bound property is a good exercise, and the principle is discussed in detail here.

Incidentally, I should admit that to the best of my knowledge real induction is really a curiosity, but it's a super cool one so I can't resist plugging it.

And in moving from $\mathbb{R}$ to $\mathbb{C}$, we lose the kind of "topological induction" above since $\mathbb{C}$ isn't linearly ordered anymore. We can still salvage it, though, via a kind of "double topological induction." However, I think it's a good idea to stop at this point.

On the other hand, at this point you can go through and cook up your own induction(-style) principles for various different mathematical structures. The most important one by far, incidentally, is induction on well-founded trees; but $\mathbb{N}$ is really the only "direct overlap" between the world of well-founded trees and the more algebraic world, so I'm not going to discuss it here.