A different answer from the ones so far: Quantum randomness is another kind of randomness that is a generalization of traditional randomness, i.e. classical or non-quantum probability. I think that it fits the question because you could likewise say that non-Euclidean geometry, interpreted as not-necessarily-Euclidean geometry, is a generalization of Euclidean geometry.
A classical probability space is usually defined as a $\sigma$-algebra $\Omega$ with a normalized measure. From the Bayesian viewpoint the measure could equally well be called a "state". Now, a $\sigma$-algebra is the algebra of Boolean random variables with a certain set of axioms. But you can just as well write down axioms for $L^\infty(\Omega)$, the algebra of bounded complex random variables. In favorable cases, it is a commutative von Neumann algebra. In quantum probability you instead allow a non-commutative von Neumann algebra $\mathcal{M}$. Also, in standard quantum probability you keep the usual completed tensor product $\mathcal{M} \otimes \mathcal{N}$ as the model of a joint system. (Free probability theory is still quantum probability, but with a certain free product instead of a tensor product.) You also still have states, conditional states, joint states, correlations, generalized stochastic maps, etc.
Some of the variant models mentioned so far lead to different theorems, but generally give the same answers in combinatorial probability, questions like the birthday paradox or modeling games of chance. Quantum probability leads to a significantly different picture of combinatorial probability, generalizing the old one, but also allowing new answers such as violation of Bell's inequalities, covariance matrices that are Hermitian rather than real symmetric, new complexity classes such as BQP, etc.
Other variant models mentioned so far no longer give any answers for combinatorial probability, for instance models of forcing. But, part of the interest in probability is that it models real life. Amazingly, so does quantum probability; that was the central discovery of quantum mechanics when it was defined in the 1920s and 1930s.
Moore and Seiberg's result (Phys. Lett. 212B (1988) p.451) on classifying modular functors can be thought of as classification of (1,2,3) theories. (M&S only do the 1 and 2 of (1,2,3), but it's not hard to extend to 3 as well; see "On Witten's 3-manifold Invariants" here.)
My guess is that extending this style of classification to any of the adjacent slots (1,2,3,4), (2,3,4) or (2,3) would be very difficult. For (1,2,3,4) one would need to start by describing a categorified action of mapping class groups of surfaces in terms of local data; the uncategorified version is already long and messy (see refs above). For (2,3,4) one would need to characterize mapping class groups of 3-manifolds in terms of local data (Hatcher-Thurston for 3-manifolds).
Best Answer
http://en.wikipedia.org/wiki/Ordered_pair#Morse_definition
Definition:
A relation $R$ is functional if and only if for all ordered pairs $\langle x,z\rangle$ and $\langle y,w\rangle$ in $R$, if $x=y$ then $z=w$.
Definition: If $R$ is a relation, $\operatorname{Range}(R) = \{y : (\exists x)(\langle x,y\rangle \in R)\}$.
Definition: A map is an ordered pair $\langle R,C\rangle$ such that $R$ is a functional relation and $\operatorname{Range}(R) \subseteq C$.