An equality type is a type to which PolymorphicEquality can be applied. The Definition and the Basis Library precisely spell out which types are equality types.

bool, char, IntInf.int, Int<N>.int, string, and Word<N>.word are equality types.

for any t, both t array and t ref are equality types.

if t is an equality type, then t list, and t vector are equality types.

if t1, …, tn are equality types, then t1 * ... * tn and {l1: t1, ..., ln: tn} are equality types.

if t1, …, tn are equality types and t AdmitsEquality, then (t1, ..., tn) t is an equality type.
To check that a type t is an equality type, use the following idiom.
structure S: sig eqtype t end =
struct
type t = ...
end
Notably, exn and real are not equality types. Neither is t1 > t2, for any t1 and t2.
Equality on arrays and ref cells is by identity, not structure. For example, ref 13 = ref 13 is false. On the other hand, equality for lists, strings, and vectors is by structure, not identity. For example, the following equalities hold.
val _ = [1, 2, 3] = 1 :: [2, 3]
val _ = "foo" = concat ["f", "o", "o"]
val _ = Vector.fromList [1, 2, 3] = Vector.tabulate (3, fn i => i + 1)