$p$-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring $A$ of characteristic $p$ a complete DVR of characteristic $0$ with residue ring $A$ generalizing $\mathbb{Z}_p$ and $\mathbb{F}_p$.
What is interesting/useful about other flavors of Witt rings, in particular the big Witt ring?
I am particularly interested in knowing their original motivation and applications.
Here is a long article by Hazewinkel and a discussion on the nLab.
The functor of taking big Witt vectors is right adjoint to the forgetful functor from lambda-rings to commutative rings. Lambda rings appear in topological K-theory and representation theory, because vector bundles (and representations of groups) have exterior powers (these are the lambda operations).
W(Z) is apparently universal in several settings, such as symmetric function theory (where I believe it is K0(finite sets)). Borger has a proposal for the category of schemes over the field with one element involving W(Z)-schemes.
I am particularly interested in knowing their original motivation and applications.
Kummer theory says that every degree-$n$ cyclic extension $L|k$ of any field $k$ containing a primitive $n$-th root $\zeta$ of $1$ is of the form $L=k(\root n\of D)$ for some order-$n$ cyclic subgroup $D\subset k^\times/k^{\times n}$, and conversely.
(Something can be said even when $\zeta\notin k$ but when $n$ is invertible in $k$. Look up a certain exercise in Schoof's book on Catalan's Conjecture.)
This leaves out degree-$p$ cyclic extensions $L|k$ of a characteristic-$p$ field. Artin-Schreier proved that $L=k(\wp^{-1}(D))$ for some ${\mathbb F}_p$-line $D\subset k/\wp(k)$, where $\wp:k\to k$ is the endomorphism $x\mapsto x^p-x$ of the additive group $k$, and conversely.
What about degree-$p^m$ cyclic extensions $L|k$ of a characteristic-$p$ field ? Many complicated constructions for particular cases were given in the 1930s (by people such as Albert) before Witt introduced the ring $W_m(k)$ of $p$-typical Witt vectors of length $m$ and the endomorphism $\wp:W_m(k)\to W_m(k)$ of the additive group, and proved that $L=k(\wp^{-1}(D))$ for some order-$p^m$ cyclic subgroup $D\subset W_m(k)/\wp(W_m(k))$, and conversely.
There were many other papers in the same volume of Crelle 176 (1937) applying Witt vectors to other outstanding problems. My favourite is Hasse's characterisation of those $\alpha$ in a finite extension $K$ of ${\mathbb Q}_p$ containing a primitive $p^m$-th root of $1$ for which the extension $K(\root p^m\of\alpha)|K$ is unramified ($p^m$-primary numbers; see for example the book by Fesenko and Vostokov, freely available on Fesenko's homepage).
See also Harder, Wittvektoren, Jahresber. Deutsch. Math.-Verein. 99 (1997), no. 1, 18--48.
An English translation of this paper has appeared in Ernst Witt, Gesammelte Abhandlungen, Springer, Berlin, 1996.
From my point of view (which is I should say basically my, probably incorrect, interpretation of Borger's) part of the interest in the big Witt ring is that, at least in the flat case where there are no issues with torsion, one does not just have Frobenius lifts at every prime - one has compatible Frobenius lifts. By compatible I mean that they commute.
This is more related to \Lambda-rings in general, but as an example consider putting a \Lambda-structure on Z[x], i.e. the affine line over the integers. Then the restriction that we want commuting Frobenius lifts means that there are precisely two \Lambda structures on Z[x] namely the obvious one where x -> x^p for each prime p and a second given by Chebyshev polynomials. This is I think clearly no longer true if we drop the commutativity condition.
To expand slightly on what Scott mentioned and to connect this back to W(-) let me say the following. From Borger's point of view the functor taking big Witt vectors should be viewed as "base forgetting" from Z to F_ 1 or in other words a \Lambda-structure should be viewed as descent data to F_ 1. From this point of view having compatible information at each prime seems, at least to me, quite natural.
I believe there are other reasons related to big de-Rham Witt cohomology - namely that if one throws away primes one loses information but I don't really know enough about it to say anything with certainty.
Other answers addressed the Witt vector functor as a right adjoint. However, it is interesting that, in equivariant homotopy theory, it can also be realized as a left adjoint.
More precisely, let $G$ be a finite group. Then there is a forgetful functor from the category of $G$-Tambara functors to the category of commutative rings with a $G$-action. This functor admits a left adjoint $L$.
Let $R$ be a commutative ring, equipped with trivial $G$-action. Then the commutative ring $W_G(R):=L(R)^G$ is the ring of $G$-typical Witt vectors. When $G=C_{p^{r-1}}$, we recover the ring of $r$-truncated $p$-typical Witt vectors. On the other hand, if we take $G=C_n$, and take limit over $n\in\mathbb N_{>0}$ ordered by division, we get the big Witt vectors (if one works with profinite groups $G$, one can simply take $G=\hat{\mathbb Z}$). A reference is Section 19 of Strickland's notes on Tambara functors.
