It is well known that in the category of all topological spaces, quotient maps aren't preserved by products (this follows from the simpler fact that $X\times (-):Top\to Top$ doesn't preserve quotients). The usual solution, if one is needed, is to change to the category of k-spaces and k-continuous maps. There are other categories where products and quotients 'get along' (e.g. $Set$, $Ab$). 


Question 1: What is a large class of categories where quotient maps are preserved by products? Topoi? (Semi)abelian categories? Categories of algebras for a monad on a given category with this property?


Now one may be only interested in a certain class of quotient maps (like surjective submersions in $Diff$, the category of finite dimensional smooth manifolds). Say, regular epimorphisms, or maps admitting local sections (assuming we're in a site), or perhaps something like surjective topological submersions, where there are sections through every point in the domain. So in this case it is not a matter of putting restrictions on $X$ such that $X\times(-)$ preserves quotients, or changing the category, but narrowing the scope of the quotient maps one wants to preserve.


Question 2: Is there are large class of quotient maps (in $Top$, or in a general category - with finite products and enough colimits) that *are* preserved by products?



Categories with products that preserve quotients
Note: I have edited this answer further because I was being silly before (unnecessarily restrictive). 


I take it you mean categories $E$ for which $- \times -: E \times E \to E$ preserves quotients. The word 'quotient' may be slightly ambiguous because sometimes people use it to mean'coequalizer', and sometimes just 'epi' (as in 'quotient object'), but I take it you mean 'coequalizer'. 


A reasonably large class would be regular categories, which includes categories of algebras of monads on $Set$ and semi-abelian categories and toposes. Here quotients = regular epis are stable under pullback and in particular are closed under taking products on either side. Furthermore, in a regular category, every quotient is a *reflexive* coequalizer, meaning a coequalizer of a pair $f, g: X \to Y$ for which there exists $h: Y \to X$ with $f \circ h = g \circ h = 1\_Y$. In particular, the two projections $\pi\_1, \pi\_2: E \to Y$ of an equivalence relation $E$ on $X$, for example the kernel pair of a quotient, form a reflexive pair by the reflexivity property. So in a regular category, where quotients = coequalizers are necessarily coequalizers of their kernel pairs, quotients are quotients of reflexive pairs. 


The reason reflexivity is relevant is a $3 \times 3$ lemma which says that in a (edit: commutative-in-parallel) diagram of $3 \times 3$ objects in which all rows and all columns are coequalizer diagrams of reflexive pairs, the diagonal is a coequalizer diagram. See the first page of Johnstone's Topos Theory. Then apply this lemma to the evident diagram whose rows are of the form 


$$X\_i \times X\_{1}' \stackrel{\to}{\to} X\_i \times X\_{2}' \to X\_i \times X\_{3}'$$ 


and whose columns are of the form 


$$X\_1 \times X\_{j}' \stackrel{\to}{\to} X\_2 \times X\_{j}' \to X\_3 \times X\_{j}'$$ 


In the category $Top$, it would therefore be natural to consider quotients by equivalence relations (or even just reflexive relations) which are preserved by taking products on each side. It's that latter condition which needs to be characterized (or at least discussed further), and I may come back to that later after I get the kids off to school. :-) 


Edit: For a discussion of topological quotients which are stable under taking a product on either side, see the paper by Day and Kelly, On topological quotient maps preserved by pullback or product, Math. Proc. Cam. Phil. Soc. 67 (1970), 553-558. Or google Day-Kelly maps to find out more. 


