Here are the questions:
So what's the difference between a second-order relation and a relation between objects? Is it related to second-order logic and could you explain so that a layman can understand?
A first-order logic has a domain, such as the natural numbers. Relations of first-order logic relate one or more objects of this domain. If there is only one object the relation can be thought of as describing a property or a predicate of the object. For example, the term R(a) means that the object with the name a has the property (or predicate) R. If R is interpreted as "is an even number" and a is interpreted as "2", then R(a) means the proposition "2 is an even number".
There are also ways to quantify the objects of the domain using the logical symbols "there exists" (∃) and "for all" (∀). These quantifiers range over all objects in the domain and only those objects. They are called "logical symbols" because a model does not assign specific meaning to these two symbols as it would for a domain or a relation.
The quoted paragraph references identity (=), so let's consider what that means as an example.
In first-order logic with identity the logical symbol, =, is a special binary relation. Just like ∃ and ∀ the model does not have to define what this logical symbol means. The identity relation works like this for all models: Suppose a is the name of an object from the domain and b is the name of an object from the domain. If a = b then the name a refers to the very same object that the name b refers to. If that is not the case, that is, if the names a and b refer to two different objects, then a ≠ b.
A second-order logic has all that a first-order logic has (except perhaps identity), but it also allows one to quantify over the relations like R even though R is not a member of the domain.
Having the ability to quantify over all relations (or properties or predicates) allows one to eliminate the need for the identity logical symbol and use the "identity of indiscernibles" instead. Here is how Wikipedia describes that concept:
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa; to suppose two things indiscernible is to suppose the same thing under two names.
This is what the identity symbol meant in first-order logic: two names refer to the very same object. In second order logic one can quantify over all those properties or relations as well as the objects in the domain. That is, a = b can be replaced by ∀R(R(a) ↔ R(b)). For all predicates R, the object represented by a possesses R if and only if the object represented by b possesses R.
Wikipedia contributors. (2019, September 22). Identity of indiscernibles. In Wikipedia, The Free Encyclopedia. Retrieved 18:16, September 27, 2019, from https://en.wikipedia.org/w/index.php?title=Identity_of_indiscernibles&oldid=917052463