$ Int(x) = $ "x is an integer."
Int is a unary predicate symbol which I understand to mean that Int takes one input x.
I have also seen Int described as a unary relation.
Would the relation be between the domain, say all numbers and {0, 1} (0 = false, 1 = true)?
Example:
5, would map to 1 (true)
5.1, would map to 0 (false)
Yes, you can interpret predicates (as distinct from the symbols representing them that the other answer covers) as maps from elements of the domain of discourse to the set $\{0,1\}$. Saying that a predicate is a "unary" relation is to treat it as a special case that maps single elements of the domain - more general relations take pairs, triples or larger $n$-tuples of elements as inputs, but their outputs are always the binary set $\{0,1\}$ or equivalently $\{false, true\}$