According to Higher Topos Theory math/0608040 topos is
a category C which behaves like the category of sets, or (more generally) the category of sheaves of sets on a topological space.
Could one elaborate on that?
There are two concepts which both get called a topos, so it depends on who you ask. The more basic notion is that of an elementary topos, which can be characterized in several ways. The simple definition:
(A power object for A is an object P(A) such that morphisms B --> P(A) are in natural bijection with subobjects of A x B, so we could rephrase the condition "C has power objects" as "the functor Sub(A x -) is representable for every object A in C").
The issue with the simple definition is that it doesn't show you why these things are actually interesting. It turns out that a great deal follows from these axioms. For example, C also has finite colimits, exponential objects, has a representable limit-preserving functor P: C^op --> Doct where Doct the category of Heyting algebras such that if f: AxB --> A is the projection map for some objects A and B in C, then P(A) --> P(AxB) has both left and right adjoints considered as a morphism of Heyting algebras, etc etc. What the long-winded definition boils down to is "an elementary topos the the category of types in some world of intuitionistic logic." There's an incredible amount of material here; the best place to start is probably MacLane and Moerdijk's Sheaves in Geometry and Logic. The main reference work is Johnstone's as-yet-unfinished Sketches of an Elephant, but I certainly wouldn't start there.
The other major notion of topos is that of a Grothendieck topos, which is the category of sheaves of sets on some site (a site is a (decently nice) category with a structure called a Grothendieck topology which generalizes the notion of "open cover" in the category of open sets in a topological space). Grothendieck topoi are elementary topoi, but the converse is not true; Giraud's Theorem classifies precisely the conditions needed for an elementary topos to be a Grothendieck topos. Depending on your point of view, you might also look at Sheaves in Geometry and Logic for more info, or you might check out Grothendieck's SGA4 for the algebraic geometry take on things.
Let us concentrate on Grothendieck topoi. As mentioned in earlier posts, these are those topoi which arise as the category of sheaves for a category equipped with a Grothendieck topology. These are those topoi which "behave the most like sheaves of sets on a topological space". Let me try to explain to what extent Grothendieck topoi are topological in nature.
First, given a continuous map f:X->Y, it produces a geometric morphism Sh(X)->Sh(Y). If X and Y are sober, then there is a bijection between Hom(X,Y) and Hom(Sh(X),Sh(Y)) (where the later is again geometric morphisms). This means, if we restrict to sober spaces, we get a fully faithful functor Sh:SobTop->topoi. (Recall via stone duality that the category of sober spaces is equivalent to the category of locales with enough points).
More generally, if G is a topological groupoid (a groupoid object in Top), we can construct its classifying topos. This can be defined as follows: Take the enriched nerve of G to obtain a simplicial space, applying the functor "Sh" (viewing the nerve as a diagram of space) and obtain a simplicial topos. Now take the (weak) colimit of the diagram to obtain a topos BG. This topos can be described concretely as equivariant sheaves over G_0.
Geometrically, BG is a model for the topos of "small sheaves" over the topological stack associated to G. In fact, on etale topological stacks* (this include all orbifolds), we also have an equivalence between maps of stacks and geometric morphisms between their categories of sheaves, so, there is a subcategory (sub-2-category) of Grothendieck topoi which is equivalent to etale topological stacks. These Grothendieck topoi are called topological etendue.
*(over sober spaces)
It turns out that a large class of topoi can be obtained as BG for some topological groupoid. In fact, every Grothendick topos "with enough points" is equivalent to BG for some topological groupoid. The more general statement is that EVERY Grothendieck topos is equivalent to BG for some localic groupoid (a groupoid object in locales). Since locales are a model for "pointless topology", we see in some sense, every Grothendieck topos is "topological". You can make sense of the statement that every Grothendieck topos is equivalent to the category of small sheaves on a localic stack.
I would tend to think that in Charley's reply above the sentence
There are two concepts which both get called a topos, so it depends on who you ask.
is misleading. As Charley mentions himself afterwards, a Grothendieck topos (a category of sheaves) is a special case of the general notion of topos. It's not like there is a conflict in terms.
One may even consider further special cases that go further in the direction from general toposes to things that look more like topological spaces: the next step are the localic toposes, which are those that are categories of sheaves on categories of open subsets of a topological space.
This, in turn, is seen in higher topos theory (search the keywors on the nLab, I can't post more than one link here, unfortunately) as the beginning of an infinite hierarchy of n-localic oo-toposes: 0-localic toposes are like ordinary topoligcal while as the n in n-localic increases they become more and more objects in "derived geometry". Again, search the nLab for these keywords for more on that.
The entry nLab: topos provides some more details and in particular provides further links, also to material on higher toposes.
As Charley mentions, topoi have many nice properties, and since a topos is something which looks like sheaves of sets on a Grothendieck site, it should be clear why a topos theory would be useful. In his book, Lurie develops, among other things, a theory of infinity-topoi, of which perhaps the main example of interest is sheaves of topological spaces on a Grothendieck site. This case is already extremely interesting.
One of the reasons why we might want to consider "sheaves of topological spaces" is because we want to obtain a construction of tmf ("topological modular forms"), which is the "universal elliptic cohomology theory". An elliptic cohomology theory is a cohomology theory (in the sense of Eilenberg-Steenrod) which is associated to an elliptic curve. By the Brown representation theorem, a cohomology theory can be identified with a spectrum (in the sense of homotopy theory), which is a sequence of topological spaces satisfying some properties. So, following this very rough reasoning, we should have a sheaf (or maybe a presheaf, but we can sheafify) of topological spaces (or rather spectra) over the moduli stack of elliptic curves. Then we would like to take global sections of this sheaf to obtain the universal elliptic cohomology theory tmf. To make all of this precise, it helps to have a good theory of sheaves of topological spaces on a Grothendieck site, and in turn a good theory of infinity-topoi. See Lurie's paper "A survey of elliptic cohomology" for some more details on this.