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.
