In category theory, a product of two objects A and B is an object AxB and a pair of morphisms p1:AxB->A and p2:AxB->B, called projections. The object AxB, if it exists, has the property that for any object C and projections f:C->A and g:C->B, there is exactly one morphism <f,g> such that that p1<f,g> = f and p2<f,g> = g. Or, to make it a little more intuitive, <f,g> makes the following diagram commute:
f/ | \g
/ <f,g> \
| | |
V V V
A <------ AxB ------> B
(That is, any path from one object to another which follows the arrows is equal.) The uniqueness of <f,g> implies that AxB is unique up to isomorphism.
Although this definition is not particularly straightforward, a little work shows that it's identical to Cartesian product when the objects are sets and the morphisms are functions.
If the category is a partially ordered set, then a product is a greatest lower bound. (There is a morphism from A to B if and only if A <= B. The definition then says that AxB is the object which is less than or equal to both A and B, yet is greater than or equal to any other object which is <= A and B.)
Some categories don't have all products. For example, the finite category 2 (consisting of two objects 0 and 1 with just their identity morphisms) has the products 1x1 == 1 and 0x0 == 0, but no object 0x1. Products are just one example of a broader category theory concept called limits.