How does first-order logic differ from higher-order logic?

In formal logic, the primary distinction between first-order logic and higher-order logic lies in the scope of their quantifiers.12

• First-order logic, also known as the lower predicate calculus, restricts quantification to the individuals within a given universe of discourse. In this system, quantifiers such as "for all" ($\forall x$) and "there exists" ($\exists x$) range only over basic objects or individuals.12
• Higher-order logic extends this capacity by allowing quantification over higher-order entities. This includes sets of individuals, properties and relations of individuals, and even sets of sets or properties of properties. For example, second-order logic permits quantification over sets of individuals, while third-order logic allows quantification over sets of those sets.12

This separation was largely established by David Hilbert and Wilhelm Ackermann in the early 20th century to distinguish basic quantification theory from more complex logical systems. While first-order logic serves as the central, foundational area of modern logic, higher-order logics provide a more expressive framework at the cost of increased complexity.12