A semigroup is a magma whose operation is associative: (a ∗ b) ∗ c = a ∗ (b ∗ c) for all elements a, b, c. Associativity means that the order of performing operations does not matter, so expressions like a ∗ b ∗ c are unambiguous without parentheses.
A semigroup with an identity element is a monoid. The positive integers under addition form a semigroup (but not a monoid, since 0 is excluded). The set of all strings over an alphabet under concatenation is a semigroup (and a monoid, with the empty string as identity).
Semigroups are the minimal structure needed for sequential composition. In a category, the collection of endomorphisms of any object (morphisms from the object to itself) forms a monoid under composition — and forgetting the identity morphism leaves a semigroup. Associativity of composition is one of the axioms of category theory.