What are the main motivations that led to coin the definition of a well ordered set? The concept of totally ordered set is, in a sense, already rather demanding:  for a set to be totally ordered, it is required ( informally) that all its elements be  comparable. 
For which reasons mathematicians went further and felt the necessity for well-ordered sets? 
Is induction the main motivation? 
Is the set of natural numbers the only "concrete" set that led mathematicians  to this abstraction of a well ordered set " in general"? 
 A: Transfinite induction is important, but in order to use it you need to already have a well-ordering in hand that you want to induct along - and why would you have such a thing if you're not trying to prove a result that's a priori about well-orderings (which doesn't really count in terms of motivation)?
The key is transfinite recursion. The term "transfinite induction" is often used to encompass transfinite recursion as well - this is a pet peeve of mine - but they're different things: induction is a proof technique while recursion is a construction method. The idea behind transfinite recursion is that you want to build some object "step-by-step," but finitely many steps aren't going to cut it. 
One obvious motivator for this is the intuitive proof of the axiom of choice: "just keep picking elements one after the other until you've built an entire choice function." Formalizing this, we run into the well-ordering principle and transfinite recursion.
Transfinite recursion also occurs outside of logic, however. For example, the Cantor-Bendixson theorem states that any uncountable closed set $C\subseteq \mathbb{R}^n$ can be written uniquely as the disjoint union of a perfect set and a countable set. My favorite proof uses transfinite recursion along $\omega_1$: we define a sequence of sets as


*

*$C_0=C$,

*$C_\lambda=\bigcap_{\alpha<\lambda}C_\alpha$ for $\lambda$ limit, and

*$C_{\alpha+1}$ is the set of limit points of $C_\alpha$.
The second-countability of $\mathbb{R}$ tells us that for some countable $\alpha$ we have $C_\alpha=C_{\alpha+1}$, and that $C\setminus C_\alpha$ is countable for every $\alpha$; meanwhile, it's easy to check that $C_\alpha=C_{\alpha+1}$ iff $C_\alpha$ is perfect. (OK fine, this only shows existence, but uniqueness isn't hard.)
And there are other examples as well (another one from topology: the Borel sets can be defined without recourse to transfinite recursion, but the Borel hierarchy - a crucial tool for analyzing Borel sets - does require it). The point is that we often have some vague idea of building an object in stages. This requires an appropriate notion of "sequence of stages;" thinking about what we need such a sequence to satisfy, we're led to the definition of well-orderings - we then prove transfinite induction, which is the thing which tells us that transfinite recursion works (and lets us analyze constructions by transfinite recursion).

As an aside, it's also worth noting that choice plays no role in transfinite recursion per se - the relevant axiom (scheme) is replacement. The way choice appears in arguments by transfinite recursion is by setting them up in the first place: e.g. to build a non-measurable set of reals, we start by well-ordering the reals using choice; the following transfinite recursion along that well-ordering, however, is choice-free.
