Thermoelectric coolers (TECs) are solid-state cooling devices which can be used in certain applications to reduce the operating temperature of electronics or increase their heat dissipation. However, the performance of the cooler is strongly influenced by the thermal system into which it is placed, and the cooler design should be optimized for a given system. In this work, the possible benefits of a TEC implemented within a realistic thermal system are quantified. Finite thermal conductances between the cooled device and the TEC and between the TEC and the heat sink are considered. The entire problem is treated using dimensionless parameters, which reduces the number of independent parameters and enables generalized performance maps which clearly show the maximum benefit (in terms of a reduced device temperature or increased device heat dissipation) that a prescribed TEC can deliver to a particular application. The use of these dimensionless parameters also allows for optimization of TEC parameters without considering the cooler detailed design geometry.