A decidable object is the categorical rendering of the notion of a decidable set from computability theory. This corresponds to the algebraic and topological concepts separable , respectively unramified object, as pointed out by Lawvere.
An object $X$ of a coherent category $\mathcal{C}$ is called decidable if its equality relation $\Delta_X:X\to X\times X\quad ,$ is complemented, as a subobject of $X\times X$ . A morphism $f:A\to B$ is called decidable if it is a decidable object in the slice category $\mathcal{C}/B$.
For an object $X$ this means that in the internal logic of the category, it is true that “for any $x,y\in X$ , either $x=y$ or $x\neq y$”.
$0$ and $1$ are always decidable, and so is every natural numbers object $N$ in a topos. A subobject of a decidable object is decidable.
Decidable maps $f:A\to B$ in the opposite of the category of commutative rings $CommRing^{op}$ are precisely the separable $B$-algebras $A$.
Of course, in a Boolean category, every object is decidable. Conversely in a topos $\mathcal{E}$, or more generally a coherent category with a subobject classifier, every object is decidable precisely if $\mathcal{E}$ is Boolean.
In constructive mathematics, where Set is not assumed Boolean, one says that a set $X$ has decidable equality if it is a decidable object of $\Set$.
A decidable subobject simply means a complemented subobject. Again, in constructive mathematics, a decidable subobject in $\Set$ is called a decidable subset.
An object $X$ in a topos $\mathcal{E}$ is called anti-decidable if $\neg\neg (x=y)$ in the internal language of $\mathcal{E}$ holds for all $x,y\in X$. A formula $\varphi$ is called almost decidable iff $\neg\varphi\vee\neg\neg\varphi$ holds and an object $X$ is called almost decidable if $x=y$ is almost decidable for $x\in X$.
B. P. Chisala, M.-M. Mawanda, Counting Measure for Kuratowski Finite Parts and Decidability , Cah.Top.Géom.Diff.Cat. XXXII 4 (1991) pp.345-353. (pdf)
A. Carboni, G. Janelidze, Decidable (=separable) objects and morphisms in lextensive categories , JPAA 110 (1996) pp.219-240.
Peter Johnstone, Sketches of an Elephant vols. I,II, Oxford UP 2002.