There is the following analogy:
$$\begin{array}{cc} \text{frames} & – & \text{commutative rings} \\ | && | \\\text{locales} & – & \text{affines schemes}\end{array}$$
Here, $\bigvee$ is analagous to $\sum$ and $\bigwedge$ is analogous to $\prod$. The category of locales is defined as the dual of the category of frames. We may also define the category of affines schemes as the dual of the category of commutative rings, this is done for example by Toen and Vaquié in relative algebraic geometry. But usually we define affine schemes geometrically as locally ringed spaces. Also, affine schemes are the local building blocks for schemes.
This leads to the following two questions:
- Is there a geometric definition of locales?
- Are locales the local building blocks for some geometric objects?
So, the second question is about completing the following analogy:
$$\begin{array}{cc} \text{frames} & – & \text{commutative rings} \\ | && | \\\text{locales} & – & \text{affines schemes} \\ | && | \\ \text{?} & – & \text{schemes}\end{array}$$
After skimming through the nlab article on locales, I would guess that locales may be defined "geometrically" as localic toposes and that perhaps the glued geometric objects are toposes, too (which ones?). This is motivated by the fact that we have a fully faithful left adjoint functor
$$\mathrm{Sh} : \mathsf{Loc} \to \mathsf{Topos}$$
from locales to toposes given by taking sheaves, and that toposes are sometimes regarded as geometric objects. (It is noted at the nlab that locales coincide with Grothendieck $(0,1)$-toposes. But this is just a reformulation of the definitions.)
But this doesn't really look similar to the construction of the spectrum of a commutative ring. It rather looks like the fully faithful embedding from (affine) schemes into the category of cocomplete symmetric monoidal categories, given by taking quasi-coherent sheaves. And this is not how we usually define affine schemes geometrically.
Perhaps there is an embedding from the category of locales into the category of locally ringed spaces? Of course, we cannot take the associated topological space of points, since that may turn out to be empty. Perhaps there is a different construction, though.
Best Answer
First, if you haven't already you should have a look at this introductory paper by P.T. Johnstone The Art of pointless thinking which gives a lot of insight on how locale theory works.
Here are some observations which I hope will answer your questions:
1) As I said in the comment, when you glue locales together along open subspaces in a way similar to affine schemes, the objects you get aren't new, they are just bigger locales so there is no need for new objects (the missing corner is "locales" again).
2) In this picture, toposes are more like Stack/groupoid objects. The analogy is not perfect; more precisely toposes are "kinds of stacks" but do not correspond to those one consider in algebraic geometry like the Artin stacks or the Deligne Munford stacks. This point of view on toposes is I think most visible in Marta Bunge An application of descent to a classification theorem for toposes (I haven't found a freely available version) but also appears in some Paper by I. Moerdijk and of course all of this is a consequence of the amazing and famous Joyal & Tierney "An Extension of the Galois Theory of Grothendieck" (which is unfortunately not that simple to find). In this sense locales are the building blocks for toposes.
3) Locales are indeed a special case of toposes; they corresponds to the "localic toposes" which are exactly the toposes that are generated by subobjects of the terminal object. This notion of localic topos can be promoted to a notion of localic morphism and is rather important in topos theory (basic theory of this notion can be found in A.4.6. of P.T. Johnstone's Sketches of an Elephant).
(This paragraph is not meant to be formal.) To some extent the idea of topos is an extension of the idea of locale: in topos theory you are trying to do topology not with open subsets as the basic objects, but with sheaves as the basic objects. A sheaf (of sets) on a space is always obtained by gluing open subspaces together (along open subspaces) so in some sense sheaves are a generalization of open subspaces. From this perspective locales are just the toposes whose "topology" can be generated by open subspaces (the use of the word topology here is informal and has not much to do with Grothendieck topology).
4) To come back to the question of seeing locales a some sort of geometric objects. As I already said in the comment, such a general point of view of people working with locales is that locales ARE geometric objects by themselves, somehow more fundamental than topological spaces. It's not locales that are "structured topological spaces" -- it's topological spaces that are structured locales (a topological space is the data of a set of points $X$, a locale $L$, and a surjective map of locales from $X$ to $L$, so they are just locales with a specific set of points marked).
5) From the discussion in the comments I get that what troubles you is to consider as geometric an object which might have not any points; I will try to address that in the end of my answer.
First, as you probably know, there is an equivalence of categories between locales having enough points (it mean enough points to distinguish the elements of the frame) and sober topological spaces. So in a first approximation one can consider that locales without (enough) points and non-sober topological spaces are pathological objects, and that except for those pathologies, locales and topological spaces are the exact same things. That is what people were doing for quite some time (the first example of point-free locales/toposes were considered as being completely pathological objects).
It appears that there is a lot of very interesting example of locales without points and that those are not that pathological after all, but this has been realized more recently. A good example of that is that there is a sublocale of $\mathbb{R}$ which is the natural domain of definition of functions that are defined "almost everywhere": as such functions cannot be evaluated at any specific point, this locale cannot have any point. I think this example is studied in length in Alex Simpson Measures, Randomness and sublocales
6) The main interest of locale theory is to realise that topology doesn't really care about points, and to some extent works better if one does not care about points. So trying to construct faithful functor from the category of locales to the category of sets is a little bit weird from this perspective. I could have answered to your comment " It is intuitive that continuous maps map points to points" that it is also the case with locales that points are sent to points, but it's just that we don't really care about points.
In fact, before Cantorian set theory which pushed everyone in mathematics to think about everything as a set, most mathematicians were not thinking about spaces like the real numbers as being sets + a topology (in the Cantor style view of sets as a discrete objects) but really as a "continuum" not formed of "discrete points". This is the point of view that locale theory is pushing forward. But if you really want to have a faithful functor from locales to sets, which has some geometrical meaning, here is a way to get one:
If you start from a frame $A$ you can see it as a distributive lattice and attached to it its Stone spectrum Stone(A) which is the compact (in general not Hausdorff) space of prime filters of $A$. Morphisms of frames are a special case of morphisms of distributive lattices, so this produces your faithful functor and I think it is the most geometric one can come up with.
The points of the corresponding locales $Loc(A)$ are the totally prime filters of $A$: "prime filter" means if $a \cup b \in P$ then $a \in P$ or $b \in P$, while "totally prime filter" means if $ \bigcup a_i \in P$ then $\exists i, a_i \in P$. So the points of the locale form in some sense a specific subspace of $Stone(A)$.
From the localic point of view this can be promoted to a morphism of locales $Loc(A) \rightarrow Stone(A)$, and it is always the case that the locale $Loc(A)$ is a dense subspace of $Stone(A)$, but of course $Loc(A)$ can have no points. You can think of points of $Stone(A)$ as an approximation to (eventually non existing) points of $Loc(A)$. But I am not sure this picture gives the correct insight on locales: for example if you applied this to the locale corresponding to an ordinary topological space (like the real number) you will get a very big and not very natural space $Stone(A)$ which is a lot more complicated than the space you started with.