Thermodynamic equilibrium (TE) is the state of a thermodynamic system (e.g., a gas cloud) such that there is no net heat flow, e.g., everything in it being the same temperature. It is a theoretical state, but is useful in modeling, e.g., the limit of "if there were less heat flow, what conditions would be approached".
Local thermodynamic equilibrium (LTE) is state where conditions in region(s) within the system are such that the temperature is virtually the same as far as a particle or photon is likely to reach in a single hop. Given the ease of modeling thermodynamic equilibrium, presuming LTE can simplify models such as those for radiative transfer, making them tractable.
Thermodynamic equilibrium (or local thermodynamic equilibrium) is the condition that produces black-body radiation and the internal characteristics of a black body.