The category theoretical definition of stacks (as given for instance in Giraud: Cohomologie non-abélienne) allow for arbitrary categories as targets (the stack condition only involves isomorphisms however). A natural example is the category of quasi-coherent sheaves (which has the category of vector bundles as a subcategory). However, when you are talking about algebraic stacks (which are category theoretic stacks fulfilling extra conditions) they only involve isomorphisms. Note that given any stack restricting to isomorphisms gives a stack in groupoids. This is what one does when one considers the algebraic stack of vector bundles: Start with the stack of vector bundles with arbitrary morphisms. This is not an algebraic stack but restricting to isomorphisms gives one.
General stacks (with non-isomorphisms) are used extensively as they encode the idea of descent. This is somewhat orthogonal to algebraic stacks which try to encode the idea of a moduli problem.
Addendum: All morphisms in a descent datum are isomorphisms (this actually
follows and does not have to be assumed). However full descent means
that you can descend objects (a descent datum of objects comes from an object downstairs) but also arbitrary morphisms (a descent datum of morphisms comes from a morphisms downstairs). These two properties together can be formulated as an equivalence of categories between the category of obejcts downstairs and the category of descent data.
Addendum 1: Charles poses an interesting question. One answer can be based on the fact there seems to also be a philosophical difference between general stacks and algebraic stacks. General stacks are based on the idea that we have some objects and relations between them, the morphisms, that can be glued together over some kind of covering. Hence, usually the objects themselves are the things of main interest and the gluing condition is just an extra (though very important) condition on such objects.
Algebraic stacks on the other hand are things that themselves are glued. The relevant idea is that groupoids are a natural generalisation of equivalence relations. (One can more or less arrive at the idea of a groupoid by thinking of bereasoned equivalence relations, elements do not just happen to be equivalent but there are specific, in general several, reasons for them to be equivalent.)
Having said that, one could start with the fact that an algebraic is the stack associated to a smooth algebraic groupoid (i.e., source and target maps are smooth). This gives a candidate generalisation by just looking at smooth algebraic categories instead. However, no natural examples that are not groupoids comes (at least) to my mind. I think the reason might be the above philosophical distinction.
First, here are some things about the four generalizations you mention:
Monoids don't fall into Diers' framework: By his Proposition 1.4.1 the terminal object in his framework is strict, i.e. any morphism $1 \to A$ is an isomorphism, which is definitely not the case for monoids.
I also wouldn't expect Diers' examples to be instances of Toen/Vaquie's framework in general, Diers' example 1.3.16, the category of pairs (ring, module over it), might be a counterexample. I don't know about Durov's setting.
Durov's geometry is in no obvious way an instance of Toen/Vaquie's framework. If you want to force it into that framework, this might be an idea to go after: Monads are monoids in the monoidal category of endofunctors. Commutative monads, however, are not commutative monoids in that category; indeed it doesn't even make sense to say that since the category is not symmetric monoidal. So first you have to find a symmetric monoidal ambient category in which Durov's generalized rings live. Seeing monads as Lawvere algebraic theories or as (things presented by) sketches might do the job - a commutative theory is probably exactly a sketch with an isomorphism from its tensor square. Another idea could be to consider a category of monads where morphisms are natural monad transformations together with distributive laws...
Derived algebraic geometry on the other hand is an instance of the homotopical version of Toen/Vaquie's framework, also contained in that article - see also below.
Second let me point out that there are many more generalizations of algebraic geometry than those four:
° Rings with extra structure can count as generalization, if one can endow any usual ring with such an extra structure, e.g.
Not Borger's geometry with lambda-rings: Not any ring can be endowed with the trivial lambda-ring structure - see his comment
Berkovich's analytic geometry: Any ring can be endowed with the trivial metric
° One can replace rings by first order structures in several ways:
Several Russian authors do this in somewhat similar ways, a recent reference is this one by Daniyarova, Myasnikov, Remeslennikov which has many references to other work in this direction; see also this one by B. Plotkin.
First order structures can be described by sketches and there is an outline of algebraic geometry along this line in this text by R. Guitart.
° There are hyperrings (used for algebraic geometry by Connes/Consani) and fuzzy rings (by Walter Wenzel and Andreas Dress, e.g. this), which are certain second order structures.
° There are two generalizations of rings used by Shai Haran to compactify the integers, F-rings and the one given in his "Non-additive prolegomena".
° There is another generalization made by Shai Haran in his article on "dyslectic geometry" in which rings are endowed with gradings over general monoids (see here).
Something quite similar seems to be going on in James Dolan's generalization of algebraic geometry (unpublished, but see here, there also was a series of videos of talks somewhere)
° derived algebraic geometry doesn't necessarily have to be based on simplicial rings; dg-algebras and $E_\infty$-ring spectra are equally important inputs, and there are many others, captured in
Toen-Vezzosi's HAG-contexts (these are homotopically additive)
Lurie's structured spaces from DAG 5 which capture about everything based on the idea of glueing together homotopical algebraic structure.
° note that Toen/Vaquie in their relative geometry do not stop at commutative monoids in some monoidal category but also give a homotopical version - this is something like a non-additive version of HAG-contexts and covers e.g. geometry over the "spectrum with one element" whose input are simplicial monoids.
° replacing rings by groupoid objects in rings together with an appropriate notion of equivalence gives you stack theory. Of course you can go on to higher stacks.
° of course there are the several approaches to non-commutative algebraic geometry - see Mahanta's survey for some of them, many related to the next point
° Chirvasitu/Johnson-Freyd's 2-schemes
° Takagi's generalized schemes
° Deitmar's congruence schemes
° Lorscheid's blue schemes
° I am sure I forgot several things...
Third, since I am at it, let me note that there are also generalizations of algebraic geometry which do not exactly build upon a generalized notion of ring, e.g.
° Hrushovski/Zilber's Zariski Geometries (see here) capturing the essential structure which is used in the applications of model theory to arithmetic geometry
° Rosenberg's noncommutative geometry. It doesn't have to be non-commutative, and also not additive, as Z. Skoda pointed out here
° in particular: schemes as dg-categories (Kontsevich, Rosenberg, Tabuada)
° Balmer's triangulated geometry
...
To summarize: Carefully mapping AG-land will keep you busy for a while. I have gathered quite some material for a rudimentary map (or maybe a low resolution satellite photo) accompanied by a few selected closer snapshots, but I won't start writing it before the second half of the year...
Best Answer
Let me try to address the bulleted questions and simultaneously advertise the G-R book everyone has mentioned. Since the main question was about literature, I could also mention Drinfeld's article "DG quotients of DG categories," which nicely summarizes the state of the general theory before $\infty$-categories shook everything up. However, it doesn't contain any algebraic geometry.
If $X = \text{Spec } A$ is an affine scheme, it's reasonable to define the category of quasicoherent sheaves $\text{QCoh}(X) := A\text{-mod}$ as the category of $A$-modules. Any other definition (e.g. via Zariski sheaves) must reproduce this answer anyway. If we understand this as the derived category of $A$-modules, then there is a canonical DG model: the homotopically projective complexes in the sense of Drinfeld's article.
The next step is to construct $\text{QCoh}(X)$ for $X$ not necessarily affine. So write $X = \cup_i \text{Spec } A_i$ as a union of open affines (say $X$ is separated to simplify things). It would be great if we could just "glue" the categories $A_i\text{-mod}$, the way that we compute global sections of a sheaf as a certain equalizer. Concretely, a complex of sheaves on $X$ should consist of complexes of $A_i$-modules for all $i$, identified on overlaps via isomorphisms satisfying cocycle "conditions" (really extra data). This is the kind of thing that totally fails in the triangulated world: limits of 1-categories just don't do the trick. Even if we work with the DG enhancements, DG categories do not form a DG category, so this doesn't help.
As you might have guessed, this is where $\infty$-categories come to the rescue. Let me gloss over details and just say that there is a (stable, $k$-linear) $\infty$-category attached to a DG category such as $A$-mod, called its DG nerve. If we take the aforementioned equalizer in the $\infty$-category of $\infty$-categories, then we do get the correct $\infty$-category $\text{QCoh}(X)$, in the sense that its homotopy category is the usual derived category of quasicoherent sheaves on $X$. (Edit: As Rune Haugseng explains in the comments, it's actually necessary to take the limit of the diagram of $\infty$-categories you get by applying $\text{QCoh}$ to the Cech nerve of the covering. The equalizer is a truncated version of this.)
But, you might be thinking, I could have just constructed a DG model for $\text{QCoh}(X)$ using injective complexes of Zariski sheaves or something. That's true, and obviously suffices for tons of applications, but as soon as you want to work with more general objects than schemes you're hosed. True, there are workarounds using DG categories for Artin stacks, but the theory gets very technical very fast.
If we instead accept the inevitability of $\infty$-categories, we can make the following bold construction. A prestack is an arbitrary functor from affine schemes to $\infty$-groupoids (i.e. spaces in the sense of homotopy theory). For example, affine schemes are representable prestacks, but prestacks also include arbitrary schemes and Artin stacks. Then for any prestack $\mathscr{X}$ we can define $\text{QCoh}(\mathscr{X})$ to be the limit of the $\infty$-categories $A\text{-mod}$ over the $\infty$-category of affine schemes $\text{Spec } A$ mapping to $\mathscr{X}$. A cofinality argument for Zariski atlases shows this agrees with our previous definition for $\mathscr{X}$ a scheme.
For example, if $\mathscr{X} = \text{pt}/G$ is the classifying stack of an algebraic group $G$, then the homotopy category of $\text{QCoh}(\mathscr{X})$ is the derived category of representations of $G$. Even cooler: if $X$ is a scheme the de Rham prestack $X_{\text{dR}}$ is defined by $$\text{Map}(S,X_{\text{dR}}) := \text{Map}(S_{\text{red}},X).$$ Then, at least if $k$ has characteristic zero, our definition of $\text{QCoh}(X_{\text{dR}})$ recovers the derived category of crystals on $X$, which can be identified with $\mathscr{D}$-modules. So we put two different ``flavors" of sheaf theory on an equal footing.