There are at least three definitions of "groupoid" currently in use.
The first type of groupoid is an algebraic structure on a set with a binary operator. The only restriction on the operator is closure (i.e., applying the binary operator to two elements of a given set returns a value which is itself a member of ). Associativity, commutativity, etc., are not required (Rosenfeld 1968, pp. 88-103). A groupoid can be empty. The numbers of nonisomorphic groupoids of this type having , 2, ... elements are 1, 10, 3330, 178981952, ... (OEIS A001329), and the corresponding numbers of nonisomorphic and nonantiisomorphic groupoids are 1, 7, 1734, 89521056, ... (OEIS A001424). An associative groupoid is called a semigroup.
The second type of groupoid is, roughly, a category which is "group-like" in the sense that every morphism (or arrow) is invertible. To make this notion more precise, one says that a groupoid is a category consisting of a set of objects and a set of arrows whereby each arrow in has an inverse arrow (also in ) subject to the identities and . Here, denotes the source of the arrow , denotes the target of , and equals the identity arrow of an object . This notion of groupoid has become widely applied throughout modern mathematics and is often seen to generalize many group-theoretic notions in a number of fields; in particular, one can define the fundamental groupoid of a manifold , as well as more general objects such as Lie groupoids, holonomy groupoids, Étale groupoids, etc. (Moerdijk and Mrčun 2003).
The third type of groupoid is an algebraic structure first defined by Brandt (1926) and also known as a virtual group. A groupoid with base (or "over ") is a set with mappings and from onto and a partially defined binary operation , satisfying the following four conditions:
1. is defined whenever , and in this case and .
2. Associativity: if either of and are defined so is the other and they are equal.
3. For each , there are left- and right-identity elements and respectively, satisfying .
4. Each has an inverse satisfying and .
Any group is a groupoid with base a single point.
The most basic example of groupoids with base is the pair groupoid, where , and , , and with multiplication . Any equivalence relation on defines a subgroupoid of the pair groupoid.
A useful way to think of a groupoid of the third type is as a parametrized equivalence relation on , as follows. Given a groupoid over , define an equivalence relation on by for each . This equivalence relation is "parameterized" because there may be more than one element in which give rise to the same equivalence,that is, and such that and .
Though it is not obvious, one can show with a bit of work that the second and third definitions are actually equivalent.