Depending on the operations and axioms, the algebraic structures get their names. The following is a partial list of algebraic structures:

- Magma or groupoid: a set with a single binary operation
- Quasigroup: a magma in which division is always possible
- Loop: a quasigroup with an identity element
- Semigroup: an associative magma
- Monoid: a semigroup with an identity element
- Group: a monoid in which every element has an inverse, or equivalently, an associative loop
- Abelian group: a commutative group
- Ring: a set with an abelian group operation as addition, together with a monoid operation as multiplication, satisfying distributivity
- Field: a ring in which the non-zero elements form an abelian group under multiplication
- Module over a given ring
*R*: a set with an abelian group operation as addition, together with an additive unary operation of scalar multiplication for every element of*R*, with an associativity condition linking scalar multiplication to multiplication in*R* - Vector space: a module over a field
- Algebra: a module or vector space together with a bilinear operation as multiplication
- Associative algebra: an algebra whose multiplication is associative
- Commutative algebra: an associative algebra whose multiplication is commutative
- Kleene algebra: two binary operations and one unary operator, modeled on regular expressions
- Lattice: a set with two commutative, associative, idempotent operations satisfying the absorption law
- Boolean algebra: a bounded, distributive, complemented lattice
- Set: although some mathematicians would not count it, a set can itself be thought of as a degenerate algebraic structure, one that has zero operations defined on it

Algebraic structures can also be defined on sets with additional non-algebraic structures, such as topological spaces.
For example, a topological group is a topological space with a group structure such that the operations of multiplication and taking inverses are continuous; a topological group has both a topological *and* an algebraic structure.
Other common examples are topological vector spaces and Lie groups.

Every algebraic structure has its own notion of homomorphism, a function that is compatible with the given operation(s). In this way, every algebraic structure defines a category. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This category, being a concrete category, may be regarded as a category of sets with extra structure in the category-theoretic sense. Similarly, the category of topological groups (with continuous group homomorphisms as morphisms) is a category of topological spaces with extra structure.\n