Local Thermodynamic Equilibrium is a condition, e.g., of some gas, where the change in Temperature is small enough that it can be considered constant, i.e., in Thermodynamic Equilibrium (TE). If processes under study directly affect matter only within sufficiently short distances, then assuming a constant temperature can be a useful approximation. The term is also used for the model that assumes the condition.
The term NLTE (for Not or Non Local Thermodynamic Equilibrium) is used for an amended LTE-like model that accommodates a common discrepancy from true LTE, specifically, excess Electromagnetic Radiation (EMR) invading the locality. Like LTE, it is an approximation used to make stellar models (and other models) tractable. The terms Kinetic Equilibrium and Statistical Equilibrium are sometimes used as an alternative to "NLTE" since the term NLTE might literally be taken to mean "global Thermodynamic Equilibrium" or to mean any other concept that isn't LTE.
Thermodynamic Equilibrium (TE)