Allegories in easy words? 
1) What is, in easy words, the definiton of an allegory?
2) And when are allegories useful?

What does it have to do with the category theory and categories?
With the definiton of category, it is easy to have an idea of what is a category, but with allegories I'm totally lost.
 A: An allegory is a special kind of category that has properties like the category $\textbf{Rel}$ of sets and relations. More precisely,
Definition. An allegory is a category $\mathcal{A}$ equipped with a poset structure on each hom set $\mathcal{A} (X, Y)$ and an identity-on-objects functor $(-)^{\circ} : \mathcal{A}^{\textrm{op}} \to \mathcal{A}$ such that 


*

*composition $\mathcal{A} (Y, Z) \times \mathcal{A} (X, Y) \to \mathcal{A} (X, Z)$ is monotonic,

*the map $(-)^{\circ} : \mathcal{A} (X, Y) \to \mathcal{A} (Y, X)$ is monotonic,

*the hom-poset $\mathcal{A}(X, Y)$ has binary meets, and

*for all $\psi : X \to Y$, $\psi : Y \to Z$ and $\chi : X \to Z$ in $\mathcal{A}$, we have the following inequality:
$$(\psi \circ \phi) \cap \chi \le (\psi \cap (\chi \circ \phi^{\circ})) \circ \phi$$
Example. The category $\textbf{Rel}$ is an allegory, where $\phi \le \phi'$ means $\phi \subseteq \phi'$ (as subsets of $X \times Y$) and $\phi^{\circ}$ is the relation $Y \to X$ such that $\phi^{\circ} (y, x)$ if and only if $\phi (x, y)$. It is clear that the poset $\textbf{Rel} (X, Y) = \mathscr{P} (X \times Y)$ has binary meets (= intersections) and so we just have to verify the displayed inequality. But, if there are $x, y, z$ such that $\chi (x, z)$, $\phi (x, y)$, and $\psi (y, z)$, then certainly $\phi (x, y)$, $\phi^{\circ} (y, x)$, $\chi (x, z)$, and $\psi (y, z)$, so the inequality is indeed true.
In fact, the same kind of reasoning shows that for any first-order theory $\mathbb{T}$ we can construct an allegory whose objects are predicates over $\mathbb{T}$ (written as $\{ (x_1, \ldots, x_n) : \alpha \}$, where $x_1, \ldots, x_n$ are the free variables of a formula $\alpha$) modulo equivalence under renaming (but not permutation) of variables, and whose arrows $\phi : \{ (x_1, \ldots, x_n) : \alpha \} \to \{ (y_1, \ldots, y_m) : \beta \}$ are those formulae $\phi$ with free variables $x_1, \ldots, x_n, y_1, \ldots, y_m$ such that $\mathbb{T}, \phi \vdash \alpha$ and $\mathbb{T}, \phi \vdash \beta$, modulo provable-equivalence under $\mathbb{T}$. Composition is, of course, the composition of relations: so given $\phi : \{ (x_1, \ldots, x_n) : \alpha \} \to \{ (y_1, \ldots, y_m) : \beta \}$ and $\psi : \{ (y_1, \ldots, y_m) : \beta \} \to \{ (z_1, \ldots, z_l) : \gamma \}$, $\psi \circ \phi$ is the predicate corresponding to $\exists y_1 . \ldots . \exists y_m . \phi \land \psi$. This is called the syntactic allegory of $\mathbb{T}$.
