A transferable utility game is described by two objects: A set $\nn$ of players and its characteristic function $v:2^\nn\to\R$. A subset of $\nn$ is called a coalition. The domain of $v$ is the set of all coalitions, including both $\nn$ and $\emptyset.$ $\nu(S)$, the total amount of utility the members of $S$ could achieve on their own, is called the value of $S$.
Cooperative game theory is concerned with the stability of utility payoffs. Can a coalition leave the game and do better on their own? If not, it must be worthwhile for the coalition to stay in. Thus cooperative game theory is concerned with distributions of utility. Details of how the game is played, strategies, equilibria, etc., disappear in this very high-level view of game outcomes.
The ideas of TU cooperative game theory can be extended to cooperative games with transferable utility. In these games the range of the characteristic function is sets of real numbers corresponding to the utility each player can achieve in a coalition. When utility is transferable, these sets are half-spaces, and there is no loss of generality in summing up the utilities on the boundary to get a TU characteristic function.