What is a category of locales? What is a locale (category of locales) and how this concept relates to the concept of a frame?
Could you explain it without advanced math (just some real analysis,  functional analysis and basic abstract algebra)?
 A: The concept of locale is a generalisation of topological space in the context of point-less topology. In other words, one realises that points are not really needed in the essential definition of topology, and that one can actually forget about them at all, if one wants to concentrate only on the intrinsic significance of topology. 
Definition. A locale $\mathfrak{X}$ is a complete Heyting algebra, namely it consists of a partially ordered set $(X,\leq)$ with two operations:
$$ \vee : \prod_{i\in I} X \longrightarrow X$$
for any set $I$ and
$$ \wedge : X\times X\longrightarrow X$$
such that


*

*$(X,\wedge,\vee)$ is a comlete lattice: for each $a,b\in X$ we have $a\wedge b \leq a,b$ and $a,b\leq a\vee b$;

*the infinite distributive law holds, namely for each $a\in X$
$$ a\wedge \bigvee_{i\in I} b_i =\bigvee_{i\in I}(a\wedge b_i)$$
where $I$ is any set and $b_i\in X$ for each $i\in I$. 
This definition mimics that of topology: indeed, every topology over a set $T$ is a locale, just by setting $\mathfrak{X}:=(\mathrm{Op}(T),\subseteq)$ and $\wedge :=\cap,\vee:= \cup$. 
Locales form a category, which means that we need to describe morphisms between locales too. Indeed, if $\mathfrak{X},\mathfrak{Y}$ are two locales, then a morphism of locales $\varphi :\mathfrak{X}\longrightarrow \mathfrak{Y}$ consists of a morphism of partially ordered sets $\varphi :(Y,\leq_Y)\longrightarrow (X,\leq_X)$, such that $$\varphi(a\wedge_Y b)=f(a)\wedge_Y f(b)$$ and
$$f\left (\bigvee_{i\in I} b_i \right )=\bigvee_{i\in I}f(b_i)$$
Note that, given  topological spaces $T,S$ and the associated locales $\mathfrak{X}_T,\mathfrak{X}_S$ of open sets, then a continuous function $f:X\longrightarrow Y$ gives rise to a morphism of locales $\varphi:\mathfrak{X}_T\longrightarrow \mathfrak{X}_S$ by setting, for each $U\in \mathfrak{X}_S$,
$$\varphi (U):=f^{-1}(U)$$
The other way around is equally true, if the locales can be turned into topological spaces. 
Frames. Let us call $\mathbf{Loc}$ the category of locales. A frame is an object of the opposite category of $\mathbf{Loc}$. This means that a frame is still a comlete Heyting algebra like a locale, but there is a tricky change to be made in morphisms. This time, a morphism of frames $\varphi:\mathfrak{X}\longrightarrow \mathfrak{Y}$ is the same thing as a morphism of locales $\varphi^\circ :\mathfrak{Y}\longrightarrow \mathfrak{X}$.
